Education  Library 


ELEMENTS 


OF 


PLANE    AND    SPHERICAL 


TRIGONOMETRY, 


WITH  THEIR   APPLICATIONS   TO 


MENSURATION,  SURVEYING,  AND 
NAVIGATION. 


BY    ELIAS    LOOMIS,    LL.D., 
if 

PKOPESSOS    OF    NATURAL   PHILOSOPHY   AND    A8TEONOMT   I1C  TALE    COLLEGE,  AND    AUTIIOE 
A  u  COCESE  OF   MAT1IEMATICS." 


TWENTY- FIFTH     EDITION. 


NEW    YORK: 
HARPER    &    BROTHERS,    PUBLISHERS, 

FRAXKLIN    SQUARE. 

1873. 


<= 


* 


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ELEMENTARY  ARITHMETIC.    166  pp.,  33,  cents. 

TREATISE  ON  ARITHMETIC.    352  pp.,  $1  03. 

ELEMENTS  OF  ALGEBRA.    Revised  Edition.    281  pp.,  $1  05. 

Key  to  Elements  of  Algebra,  for  Use  of  Teachers.     128  pp.,  $1  05. 
TREATISE  ON  ALGEBRA.    Revised  Edition.    384  pp.,  $1  17. 

Key  to  Treatise  on  Algebra,  for  Use  of  Teachers.    219  pp.,  $1  17. 
ELEMENTS  OF  GEOMETRY.    Revised  Edition.    388  pp.,  $1  17. 

ELEMENTS  OF  TRIGONOMETRY,  SURVEYING,  AND  NAVIGATION.    194  pp.,  $1  17. 
TABLES  OF  LOGARITHMS.    150  pp.,  $1  17. 

The  Trigonometry  and  Tables,  bound  in  one  volume.    360  pp.,  $1  75. 
ELEMENTS.  OF  ANALYTICAL  GEOMETRY.    Revised  Edition.    261  pp.,  $1  17. 
DIFFERENTIAL  AND  INTEGRAL  CALCULUS.    Revised  Edition.    309  pp.,$l  17. 

The  Analytical  Geometry  and  Calculus,  bound  in  one  volume.    570  pp.,  $2  05. 
ELEMENTS  OF  NATURAL  PHILOSOPHY.    351  pp.,  $1  25. 
ELEMENTS  OF  ASTRONOMY.    254  pp.,  $1  17. 
PRACTICAL  ASTRONOMY.    499  pp.,  $1  75. 
TREATISE  ON  ASTRONOMY.    351  pp.,  $1  75. 
TREATISE  ON  METEOROLOGY.    308  pp.,  $1  75. 


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PREFACE. 


*TiiE  following  treatise  constitutes  the  third  volume  of  ii 
course  of  Mathematics  designed  for  colleges  and  high  schools, 
and  is  prepared  upon  substantially  the  same  model  as  the  works 
on  Algebra  and  Geometry.  It  does  not  profess  to  embody 
every  thing  which  is  known  on  the  subject  of  Trigonometry, 
but  it  contains  those  principles  which  are  most  important  on 
account  of  their  applications,  or  their  connection  with  other 
parts  of  a  course  of  mathematical  study.  The  aim  has  been 
to  render  every  principle  intelligible,  not  by  the  repetition  of 
superfluous  words,  but  by  the  use  of  precise  and  appropriate 
language.  "Whenever  it  could  conveniently  be  done,  the  most 
important  principles  have  been  reduced  to  the  form  of  theorems 
or  rules,  which  are  distinguished  by  the  use  of  italic  letters, 
and  are  designed  to  be  committed  to  memory.  The  most  im- 
portant instruments  used  in  Surveying  are  fully  described,  and 
are  illustrated  by  drawings. 

The  computations  are  all  made  by  the  aid  of  natural  num- 
bers, or  with  logarithms  to  six  places ;  and  by  means  of  the 
accompanying  tables,  such  computations  can  be  performed 
with  great  facility  and  precision  This  volume,  having  been 
ttsed  by  several  successive  classes,  has  been  subjected  to  tho 
ssverest  scrutiny,  and  the  present  edition  embodies  all  the  al- 
terations which  have  been  suggested  by  experience  in  the  re- 
oitaHoo  zcom. 


L7 


CONTENTS, 

.  ___  _  __  i*]  U  < 

BOOK  I. 

THE  NATURE  AND  PROPERTIES  OF  LOGARITHMS. 

r»-< 

Nature  of  Logarithms  .................................................  1 

Description  of  the  Table  of  Logarithms  ..................................  9 

Multiplication  by  Logarithms  ..........................................  1  . 

Division  by  Logarithms  ...............................................  1C 

Involution  by  Logarithms  .............................................  17 

Evolution  by  Logarithms  .......  .  ......................................  17 

Proportion  by  Logarithms  .............................................  18 

BOOK  II. 

PLANE  TRIGONOMETRY. 

Kines,  Tangents,  Secants,  &c.,  defined  ..................................  20 

Explanation  of  the  Trigonometrical  Tables  ...............................  23 

To  find  Sines  and  Tangents  of  small  Arcs  ...............................  29 

Solutions  of  Right-angled  Triangles  .....................................  32 

Solutions  of  Oblique-angled  Triangles  ...................................  3G 

Instruments  used  in  Drawing  ..........................................  42 

Geometrical  Construction  of  Triangles  ..................................  46 

Values  of  the  Sines,  Cosines,  &c.,  of  certain  Angles  .......................  48 

Trigonometrical  Formula)  ..............................................  52 

Computation  of  a  Table  of  Sines,  Cosines,  &c  .............................  57 

BOOK  III. 

MENSURATION  OF  SURFACES  AND  SOLIDS. 

Areas  of  Figures  bounded  by  Right  Lines  ................................  5S> 

Area  of  a  Regular  Polygon  .............................................  64 

Quadrature  of  the  Circle  and  its  Parts  ........  :  ..........................  6G 

Mensuration  of  Solids  .................................................  71 

Rail-way  Excavations  or  Embankments  .................................  77 

Regular  Polyedrons  ...................................................  81 

The  three  Round  Bodies  ...............................................  84 

Area  of  a  Spherical  Triangle  ...........................................  8S 

BOOK  IV. 

SURVEYING. 

Definitions  ..........................................................  HG 

Instruments  for  measuring  Angles  ...........................  .  ..........  91 


ri  CONTENTS. 

+»* 

Explanation  of  the  Vernier  ............................................     94 

Description  of  the  Theodolite  .........................................     9o 

Heights  and  Distances  ...............................................     97 

The  Determination  of  Areas  ..........................................  303 

Plotting  a  Survey  ....................................  .  ...............  101 

The  Traverse  Table  ......  ............................................  1  06 

To  find  the  Area  of  a  Field  ...........................................  109 

Trigonometrical  Surveys  .....................  .....  ___  .........  .......  114 

Variation  of  the  Needle  ..............................................  117 

Leveling  ...........................................................  119 

Topographical  Maps  .................................................  123 

Setting  out  Rail-way  Curves  ..........................................  127 

Surveying  Harbors  ..................................................  130 

The  Plane  Table  ....................................................  132 

To  determine  the  Depth  of  Water  .....................................  133 

BOOK  V. 

NAVIGATION. 

Definitions,  &c  ......................................................  135 

Plane  Sailing  .......................................................  138 

Traverse  Sailing  .....................................................  141 

Parallel  Sailing  .....................................................  144 

Middle  Latitude  Sailing  ........  .  ......................................  1  46 

Mercator's  Sailing  ...................................................  149 

Nautical  Charts  .....................................................  J  53 

BOOK  VI. 

SPHERICAL  TRIGONOMETRY. 
Right-angled  Spherical  Triangles 


Napier's  Rule  of  the  Circular  Parts  ....................................  158 

Examples  of  Right-angled  Triangles  ...................................  160 

Oblique-angled  Spherical  Triangles  ....................................  163 

Examples  of  Oblique-angled  Triangles  .............  ............  ........  165 

Trigonometrical  Formulae  ..........  ..................................  171 

Bailing  'on  an  Arc  of  a  Great  Circle  ....................................  17« 


TElGOIOMETliY. 


BOOK  I. 

THE  NATURE  AND  PROPERTIES  OF  LOGARITHMS. 

ARTICLE  L  Logarithms  are  numbers  designed  to  dimmieii 
th  3  labor  of  Multiplication  and  Division,  by  substituting  in  their 
stead  Addition  and  Subtraction.  All  numbers  are  regarded  a* 

O 

powers  of  some  one  number,  which  is  called  the  base  of  the 
•system  ;  and  the  exponent  of  that  power  of  the  base  which  is 
equal  to  a  given  number,  is  called  the  logarithm  of  that  number. 

The  base  of  the  common  system  of  logarithms  (called,  from 
their  inventor,  Briggs'  logarithms)  is  the  number  10.  Hence 
all  numbers  are  to  be  regarded  as  powers  of  10.  Thus,  since 
10°— 1,  0  is  the  logarithm  of  1  in  Briggs'  system; 

10'  =  10,         1      "  "  10  «  " 

105=100,       2      "  "  100  "  «' 

103=1000,     3      "  "  1000  "  « 

104=10000,  4      "  "  10,000  «  " 

&c.,  &c.,  &c. ; 

whence  it  appears  that,  in  Briggs'  system,  the  logarithm  oi 
every  number  between  1  and  10  is  some  number  between  0 
and  1,  i.  e.,  is  a  proper  fraction.  The  logarithm  of  every  num- 
ber between  10  and  100  is  some  number  between  1  and  2,  i.  e.j 
is  1  plus  a  fraction.  The  logarithm  of  every  number  between 
100  and  1000  is  some  number  between  2  and  3,  i.  e.,  is  2  plus 
a  fraction,  and  so  on. 

(2.)  The  preceding  principles  may  be  extended  to  fractions 
by  means  of  negative  exponents.     Thus,  since 
10-'= 0.1,        —1  is  the  logarithm  of  0.1       in  Briggs'  system ; 
10-2=0.01,      -2     "  «  0.01  «  " 

10-' =0.001,    -3     "  "  0.001  "  " 

U)-4=0.0001  -4     "  ••  0.0001          «  « 


H  TRIGONOMETRY. 

IIciioc  it  appears  that  the  logarithm  of  every  number  between 
1  and  0  1  is  some  number  between  0  and  —  1,  or  may  be  rep 
resented  by  —1  plus  a  fraction  ;  the  logarithm  of  every  num- 
ber between  0.1  and  .01  is  some  number  between  —  1  and  —2, 
or  may  be  represented  by  —2  plus  a  fraction;  the  logarithm 
of  every  number  between  .01  and  .001  is  some  number  be- 
tween —  2  and  —3,  or  is  '»nual  to  —3  plus  a  fraction,  and 
soon. 

The  logarithms  of  most  numbers,  therefore,  consist  of  an  in 
teger  and  a  fraction.  The  integral  part  is  called  the  charac 
teristic,  and  may  be  known  from  the  following 

RULE. 

The  characteristic  of  the  logarithm  of  any  number  greater 
than  unity,  is  one  less  than  the  number  of  integral  figures  in 
the  given  number. 

Thus  the  logarithm  of  297  is  2  plus  a  fraction  ;  that  is,  the 
characteristic  of  the  logarithm  of  297  is  2,  which  is  one  less 
than  the  number  of  integral  figures.  The  characteristic  of  the 
logarithm  of  5673.29  is  3  ;  that  of  73254.1  is  4,  &o. 

The  characteristic  of  the  logarithm  of  a  decimal  fraction 
is  a  negative  number,  and  is  equal  to  the  number  of  places  by 
which  its  first  significant  figure  is  removed  from  the  place 
of  units. 

Thus  the  logarithm  of  .0046  is  —3  plus  a  fraction;  that  is, 
the  characteristic  of  the  logarithm  is  —3,  the  first  significant 
figure,  4,  being  removed  three  places  from  units. 

(3.)  Since  powers  of  the  same  quantity  are  multiplied  by 
adding  their  exponents  (Alg.,  Art.  50), 

The  logarithm  of  the  product  of  two  or  more  factors  z.s 
fqual  to  the  sum  of  the  logarithms  of  those  factors. 

Hence  we  see  that  if  it  is  required  to  multiply  two  or  more 
numbers  by  each  other,  we  have  only  to  add  their  logarithms  : 
the  sum  will  be  the  logarithm  of  their  product.  "We  then  look- 
in  the  table  for  the  number  answering  to  that  logarithm,  in 
order  to  obtain  the  required  product. 

Also,  since  powers  of  the  same  quantity  are  divided  by  sub- 
tracting their  exponents  (Alg1.,  Art.  66), 

The  logarithm  of  the  quotient  of  one  number  divided  lit  an- 


LOGARITHMS.  S 

othei,  in  equal  to  the  difference  of  the  logarithms,  of  those 
numbers. 

Hence  we  see  that  if  we  wish  to  divide  one  number  by  an- 
other, we  have  only  to  subtract  the  logarithm  of  the  divisor 
from  that  of  the  dividend  ;  the  difference  will  be  the  logarithm 
3f  their  quotient. 

(4.)  Since,  in  Briggs'  system,  the  logarithm  of  10  is  1,  it 
any  number  be  multiplied  or  divided  by  10,  its  logarithm  will 
be  increased  or  diminished  by  1 ;  and  as  this  is  an  integer,  it 
will  only  change  the  characteristic  of  the  logarithm,  without 
affecting  the  decimal  part.  Hence 

The  decimal  part  of  the  logarithm  of  any  number  is  the 
same  as  that  of  the  number  multiplied  or  divided  by  10,  100, 
1000,  &c. 

Thus,  the  logarithm  of  65430  is  4.815777  , 

«  "  6543  is  3.815777; 

«  "  654.3  is  2.815777; 

«  "  65.43          is  1.815777; 

"  «  6.543        isO.S15r;77; 

"  "  ,6543      is  1.815777 ; 

"  "  .06543    is  2.815777; 

"  "  .006543  is  3.815777. 

The  minus  sign  is  here  placed  over  the  characteristic,  to 
show  that  that  alone  is  negative,  while  the  decirial  part  of  the 
logarithm  is  positive. 

TABLE  OF  LOGARITHMS. 

(5.)  A  table  of  logarithms  usually  contains  the  logarithms 
of  the  entire  series  of  natural  numbers  from  1  up  to  10,000, 
and  the  larger  tables  extend  to  100,000  or  more.  In  the  smaller 
tables  the  logarithms  are  usually  given  to  five  or  six  decimal 
places  ;  the  larger  tables  extend  to  seven,  and  sometimes  eight 
or  more  places. 

In  the  accompanying  table,  the  logarithms  of  the  first  100 
numbers  are  given  with  their  characteristics  ;  but,  for  all  other 
numbers,  the  decimal  part  only  of  the  logarithm  is  given,  whilo 
the  character stic  is  left  t>  be  supplied,  according  to  the 
in  Ait,  2. 


10  TRIGONOMETRY. 

(6.)  To  fend  Hie  Logarithm  of  any  Number  between  1  and  100 
Look  on  the  first  page  of  the  accompanying  table,  along  the 
column  of  numbers  under  N.,  for  the  given  number,  and  against 
it,  in  the  next  column,  will  be  found  the  logarithm  with  its 
characteristic.  Thus, 

opposite  13  is  1.113943,  which  is  the  logarithm  of  13 ; 
"        65  is  1.812913,  "  «  65. 

To  find  the  Logarithm  of  any  Number  consisting  of  three 

Figures. 

Look  on  one  of  the  pages  of  the  table  from  2  to  20,  alon^ 
the  left-hand  column,  marked  N.,  for  the  given  number,  and 
against  it,  in  the  column  headed  0,  will  be  found  the  decimal 
part  of  its  logarithm.  To  this  the  characteristic  must  be  pre- 
fixed, according  to  the  rule  in  Art.  2.  Thus 
the  logarithm  of  347  will  be  found,  from  page  8,  2.540329 ; 

«  "          871  "  "       18,  2.940018. 

As  the  first  two  figures  of  the  decimal  are  the  same  for  sev 
eral  successive  numbers  in  the  table,  they  are  not  repeated  for 
each  logarithm  separately,  but  are  left  to  be  supplied.  Thus 
the  decimal  part  of  the  logarithm  of  339  is  .530200.  The  first 
two  figures  of  the  decimal  remain  the  same  up  to  347 ;  they 
are  therefore  omitted  in  the  table,  and  are  to  be  supplied. 

To  find  the  Logarithm  of  any  Number  consisting  of  foiu 

Figures. 

Find  the  three  left-hand  figures  in  the  column  marKecl  JN  , 
as  before,  and  the  fourth  figure  at  the  head  of  one  of  the  other 
columns.  Opposite  to  the  first  three  figures,  and  in  the  col- 
umn under  the  fourth  figure,  will  be  found  four  figures  of  the 
logarithm,  to  which  two  figures  from  the  column  headed  0  are 
to  be  prefixed,  as  in  the  former  case.  The  characteristic  must 
be  supplied  according  to  Art.  2.  Thus 

the  logarithm  of  3456  is  3.538574  ; 
"  "          8765  is  3.942752. 

In  several  of  the  columns  headed  1,  2,  3,  &c.,  small  dots  are 
tound  in  the  place  of  figures.  This  is  to  show  that  the  two 
figures  which  are  to  be  prefixed  from  the  first  column  have 
ahangod.  and  thn,y  are  to  be  taken  from  the  horizrntal  line  di- 


LOGARITHMS.  (1 

reefcly  Idow.     The  place  of  the  dots  is  to  "be  supplied  with  oi 
phers.     Thus 

the  logarithm  of  2045  is  3.310693  ; 
"  "  9777  is  3.990206. 

The  two  leading  figures  from  the  column  0  must  also  be 
taken  f.*om  the  horizontal  line  below,  if  any  dots  have  "been 
oassed  over  on  the  same  horizontal  line.     Thus 
the  logarithm  of  1628  is  3.211654. 

To  find  the  'logarithm  of  any  Number  containing'  more  than 

four  Figures. 

(7.)  By  inspecting  the  table,  we  shall  find  that,  within  cer- 
tain limits,  the  differences  of  the  logarithms  are  nearly  propor- 
tional to  the  differences  of  their  corresponding  numbers.    Thus 
the  logarithm  of  7250  is  3.860338  ; 
"  "  7251  is  3.860398 ; 

"  "  7252  is  3.860458 ; 

"  "  7253  is  3.860518. 

Here  the  difference  between  the  successive  logarithms,  called 
the  tabular  difference,  is  constantly  60,  corresponding  to  a  dif- 
ference of  unity  in  the  natural  numbers.  If,  then,  we  sup- 
pose the  logarithms  to  be  proportional  to  their  corresponding 
numbers  (as  they  are  nearly),  a  difference  of  0.1  in  the  num- 
bers should  correspond  to  a  difference  of  6  in  the  logarithms , 
a  difference  of  0.2  in  the  numbers  should  correspond  to  a  dif- 
ference of  12  in  the  logarithms,  &c.  Hence 

the  logarithm  of  7250.1  must  be  3.860344  ; 
«  "          7250.2         "       3.860350; 

"  "          7250.3         "      3.860356. 

In  order  to  facilitate  the  computation,  the  tabular  difterenco 
is  inserted  on  page  16  in  the  column  headed  D.,  and  the  pro- 
portional part  for  the  fifth  figure  of  the  natural  number  is  given 
at  the  bottom  of  the  page.  Thus,  when  the  tabular  difference 
is  60,  the  corrections  for  .1,  .2,  .3,  &c.,  are  seen  to  be  6,  12, 
18,  &c. 

If  the  given  number  was  72501,  the  characteristic  of  its  log. 
arithm  would  be  4,  but  the  decimal  part  would  be  the  same  as 
for  7250,1. 

If  it  were  required  to  find  the  correction  for  a  sixth  fissure 


12  TRIGONOMETRY. 

m  the  natural  number,  it  is  readily  obtained  from  the  Propor- 
tional Parts  in  the  table.  The  correction  for  a  figure  in  tha 
sixth  place  must  be  one  tenth  of  the  correction  for  the  samo 
figure  if  it  stood  in  the  fifth  place.  Thus,  if  the  correction  for 
.5  is  30,  the  correction  for  .05  is  obviously  3. 

As  the  differences  change  rapidly  in  the  first  part  of  the  ta- 
Lie,  it  was  found  inconvenient  to  give  the  proportional  parts 
for  each  tabular  difference ;  accordingly,  for  the  first  seven 
pages,  they  are  only  given  for  the  even  differences,  but  the  pro- 
portional parts  for  the  odd  differences  will  be  readily  found  by 
inspection. 

Required  the  logarithm  of  452789. 

The  logarithm  of  452700  is  5.655810. 
-  The  tabular  difference  is  96. 

Accordingly,  the  correction  for  the  fifth  figure,  8,  is  77,  and 
for  the  sixth  figure,  9,  is  8.6,  or  9  nearly.  Adding  these  cor- 
rections to  the  number  before  found,  we  obtain  5.655896. 

The  preceding  logarithms  do  not  pretend  to  be  perfectly 
exact,  but  only  the  nearest  numbers  limited  to  six  decimal 
places.  Accordingly,  when  the  fraction  which  is  omitted  ex- 
ceeds half  a  unit  in  the  sixth  decimal  place,  the  last  figure 
must  be  increased  by  unity. 

Required  the  logarithm  of  8765432. 

The  logarithm  of  8765000  is  6.942752 

Correction  for  the  fifth  figure,  4,  20 

"  "       sixth  figure,  3,  1.5 

"  "       seventh  figure,  2.  0.1 

Therefore  the  logarithm  of  8765432  is  6.942774. 

Required  the  logarithm  of  234567. 

The  logarithm  of  234500  is  5.370143 

Correction  for  the  fifth  figure,  6,  111 

«  «       sixth  figure,  7,  13 

Therefore  the  logarithm  of  234567  is       5.370267. 

To  find  the  Logarithm  of  a  Decimal  Fraction 

(8.)  According  to  Art.  4,  the  decimal  part  of  the  logarithm 
of  any  number  is  the  same  as  that  of  the  number  multiplied 
9r  divided  by  10,  100,  1000,  &c.  Hence,  for  a  decimal  frac 


LOGARITHMS.  13 

lion,  we  find  the  logarithm  as  if  the  figures  were  integers,  and 
prefix  the  characteristic  according  to  the  rule  of  Art.  2 

EXAMPLES. 

The  logarithm  of  345.6  is  2.538574  ; 

«  «  87.65          is  1.942752 : 

"  "  2.345        is  0.370143; 

"  .1234      is  1.091315 ; 

"  "  .005678  is  3.754195. 

To  find  the  Logarithm  of  a  Vulgar  Fraction. 
(9.)  We  may  reduce  the  vulgar  fraction  to  a  decimal,  and 
find  its  logarithm  by  the  preceding  article ;  or,  since  the  value 
of  a  fraction  is  equal  to  the  quotient  of  the  numerator  divided 
by  the  denominator,  we  may,  according  to  Art.  3,  subtract  the 
logarithm  of  the  denominator  from  that  of  the  numerator; 
*,ho  difference  will  be  the  logarithm  of  the  fraction. 
Ex.  1.  Find  the  logarithm  of  T3F,  or  0.1875. 

From  the  logarithm  of  3,  0.477121, 

Take  the  logarithm  of  16,  1.204120. 

Leaves  the  logarithm  of  T3F,  or  .1875,  1.273001. 
Ex.  2.  The  logarithm  of  ^  is  2.861697. 
Ex.  3    The  logarithm  of  iff  is  1.147401. 

To  find  the  Natural  Number  corresponding  to  any  Logarithm. 

(10.)  Look  in  the  table,  in  the  column  headed  0,  for  the  first, 
two  figures  of  the  logarithm,  neglecting  the  characteristic  ;  the 
other  four  figures  are  to  be  looked  for  in  the  same  column,  01 
in  one  of  the  nine  following  columns  ;  and  if  they  are  exactly 
found,  the  first  three  figures  of  the  corresponding  number  will 
be  found  opposite  to  them  in  the  column  headed  N.,  and  the 
fourth  figure  will  be  found  at  the  top  of  the  page.  This  number 
must  be  made  to  correspond  with  the  characteristic  of  the  given 
logarithm  by  pointing  off  decimals  or  annexing  ciphers.  Thus 
the  natural  number  belonging  to  the  log.  4.370143  is  23450; 
"  «  «  "  1.538574  is  34.56. 

If  the  decimal  part  of  the  logarithm  can  not  be  exactly  found 
ill  the  table,  look  for  the  nearest  less  logarithm,  and  take  out 

L 


£4 

the  lour  figures  of  the  corresponding  natural  number  as  be- 
fore ;  the  additional  figures  may  be  obtained  by  means  of  the 
Proportional  Parts  at  the  bottom  of  the  page. 

Required  the  number  belonging  to  the  logarithm  4.368399. 

On  page  6,  we  find  the  next  less  logarithm  .368287 

The  four  corresponding  figures  of  the  natural  number  are 
12335.  Their  logarithm  is  less  than  the  one  proposed  by  112. 
The  tabular  difference  is  186 ;  and,  by  referring  to  the  bottom 
of  page  6,  we  find  that,  with  a  difference  of  186,  the  figure 
corresponding  to  the  proportional  part  112  is  6.  Hence  the 
five  figures  of  the  natural  number  are  23356 ;  and,  since  the 
characteristic  of  the  proposed  logarithm  is  4,  these  five  figures 
aie  all  integral. 

Required  the  number  belonging  to  logarithm  5.345678. 

The  next  less  logarithm  in  the  table  is  345570, 

Their  difference  is  108 

The  first  four  figures  of  the  natural  number  are     2216. 

With  the  tabular  difference  196,  the  fifth  figure,  correspond- 
ing to  108,  is  seen  to  be  5,  with  a  remainder  of  10.  To  find 
the  sixth  figure  corresponding  to  this  remainder  10,  we  may 
multiply  it  by  10,  making  100,  and  search  for  100  in  the  same- 
line  of  proportional  parts.  "We  see  that  a  difference  of  100 
would  give  us  5  in  the  fifth  place  of  the  natural  number. 
Therefore,  a  difference  of  10  must  give  us  5  in  the  sixth  place 
of  the  natural  number.  Hence  the  required  number  is  221655 

In  tho  same  manner  we  find 

tin  number  corresponding  to  log.  3.538672  is  3456.78 ; 

"       1.994605  is      98.7654; 
"  "  "      1.647817  is          .444444 

MULTIPLICATION  BY  LOGARITHMS. 

(11.)  According  to  Art.  3,  the  logarithm  of  the  product  oi 
two  or  more  factors  is  equal  to  the  sum  of  the  logarithms  of 
those  factors.  Hence,  for  multiplication  by  logarithms,  we 
have  the  following 

RULE. 

Add  the  logarithms  of  the  factors  ;  the  sum  will  be  the  log 
srithm  of  their  product. 

EK,  1.  Required  the  product  of  57,98  by  18. 


LOGARITHMS.  13 

The  logarithm  of  57.98          is  1.763278 
«  "  18  is  1.255273 

The  bgarithm  of  the  product  1043.64  is  3.018551 
Ex.  2.  Required  the  product  of  397.65  by  43.78. 

Ans.,  17409.117. 

Ex.  3.  Required  the  continued  product  of  54.32,  6.543,  and 
12.345. 

The  word  sum,  in  the.  preceding  rule,  is  to  be  understood  in 
its  algebraic  sense ;  therefore,  if  any  of  the  characteristics  of 
the  logarithms  are  negative,  we  must  take  the  difference  be- 
tween their  sum  and  that  of  the  positive  characteristics,  and 
prefix  the  sign  of  the  greater.  It  should  be  remembered  that 
the  decimal  part  of  the  logarithm  is  invariably  positive ;  hence- 
that  which  is  carried  from  the  decimal  part  to  the  character- 
istic must  be  considered  positive. 
Ex.  4.  Multiply  0.00563  b^  17. 

The  logarithm  of  0.00563  is  3.75050S 
17      is  1.230449 

Product,  0.09571,  whose  logarithm  is  2.980957. 

Ex.  5.  Multiply  0.3854  by  0.0576.  Ans.  0.022199. 

Ex.  6.  Multiply  0.007853  by  0.00476. 

Ans.,  0.0000373S. 

Ex.  7.  Find  the  continued  product  of  11.35,  0.072,  and  0.017 
(12.)  Negative  quantities  may  be  multiplied  by  means  of 
logarithms  in  the  same  manner  as  positive,  the  proper  sign 
being  prefixed  to  the  result  according  to  the  rules  of  Algebra. 
To  distinguish  the  negative  sign  of  a  natural  number  iron*  th« 
negative  characteristic  of  a  logarithm,  we  append  the  letVr  n 
to  the  logarithm  of  a  negative  factor.  Thus 

the  logarithm  of  -56  we  write  1.748188  n. 
Ex.  8.  Multiply  53.46  by  -29.47. 

The  logarithm  of     53.46  is  1.728029 

-29.47  is  1.469380  n. 
Product,  -1575.47,  log.  3.197409  n. 

Ex.  9.  Find  the  continued  product  of  372.1,  -.0054,  and 
-175.6. 

Ex.  10.  Find  the  continued  product  of  -0  137,  —7.689,  and 
-  0376 


Ifi  TRIGONOMETRY. 

DIVISION  BY  LOGARITHMS 

(13.)  According  to  Art.  3,  the  logarithm  of  the  quotient  of 
ono  number  divided  "by  another  is  equal  to  the  difference  of 
the  logarithms  of  those  numbers.  Hence,  for  division  "by  log. 
arithms,  we  have  the  following 

RULE. 

From  the  logarithm  of  the  dividend,  subtract  the  logarithm 
of  the  divisor ;  the  difference  will  be  the  logarithm  of  tht 
quotient. 

Ex.  1.  Required  the  quotient  v\f  888.7  divided  by  42.24 
The  logarithm  of  888.7  is  2.948755 
«  «          42.24  is  1.625724 

The  quotient  is  21.039,  whose  log.  is  1.323031. 
Ex.  2.  Required  the  quotient  of  3807.6  divided  by  13.7. 

Ans.,  277.927. 

The  word  difference,  in  the  preceding  rule,  is  to  be  under, 
tood  in  its  algebraic  sense ;  therefore,  if  the  characteristic  of 
one  of  the  logarithms  is  negative,  or  the  lower  one  is  greater 
than  the  upper,  we  must  change  the  sign  of  the  subtrahend, 
and  proceed  as  in  addition.     If  unity  is  carried  from  the  deci- 
mal part,  this  must  be  considered  as  positive,  and  must  bn 
united  with  the  characteristic  before  its  sign  is  changed. 
Ex.  3.  Required  the  quotient  of  56.4  divided  l-y  0.00015. 
The  logarithm  of        56.4  is  1.7oi279 
"  "  0.00015  is  4.176091 

The  quotient  is  376000,  whose  log.  is  5.575188. 
This  result  may  be  verified  in  the  same  way  as  subtraction 
in  common  arithmetic.  The  remainder,  added  to  the  subtra- 
hend, should  be  equal  to  the  minuend.  This  precaution  should 
always  be  observed  when  there  is  any  doubt  with  regard  to 
the  sign  of  the  result. 

Ex.  4.  Required  the  quotient  of  .8692  divided  by  42.258. 

Ans. 

Ex.  5.  Required  the  quotient  of  .74274  divided  by  .00928 
Ex.  6.  Required  the  quotient  of  24.934  divided  by  .078541 
Negative  quantities  may  be  divided  by  means  of  logarithms 


•Li  0  G  A  R  IT  II  M  S.  -17 

in  the  same  manner  as  positive,  the  pr  >per  sign  being  prefixed 
to  the  result  according  to  the  rules  of  Algebra. 

Ex.  7.  Required  the  quotient  of  -79.54  divided  by  0,08321 
Ex.  8.  Required  the  quotient  of  -0.4753  divided  by  -36.74. 

INVOLUTION  BY  LOGARITHMS. 

(14.)  It  is  proved  in  Algebra,  Art.  340,  that  the  logarithm 
of  any  power  of  a  number  is  equal  to  the  logarithm  of  that 
number  multiplied  by  the  exponent  of  the  power.  Henco,  to 
involve  a  number  by  logarithms,  we  have  the  folk  wing 

RULE. 

Multiply  the  logarithm  of  the  number  by  the  exponent  of 
the  power  required. 

Ex.  1.  Required  the  square  of  428. 

The  logarithm  of  428  is  2.631444 
2 

Square,  183184,  log.  5.262888. 
Ex.  2.  Required  the  20th  power  of  1.06. 

The  logarithm  of  1.06  is  0.025306 

20 


20th  power,  3.2071,  log.  0.506120. 

Ex.  3.  Required  the  5th  power  of  2.846. 

[t  should  be  remembered,  that  what  is  carried  from  the  dec- 
imal part  of  the  logarithm  is  positive,  whether  the  characteris- 
tic is  positive  or  negative. 

Ex.  4.  Required  the  cube  of  .07654. 

The  logarithm  of  .07654  is  2.883888 

3 

Cube,  .0004484,  log.  4.651664. 
Ex.  5.  Required  the  fourth  power  of  0.09874. 
Ex.  6.  Required  the  seventh  power  of  0.8952. 

EVOLUTION  BY  LOGARITHMS. 

(15.)  It  is  proved  in  Algebra,  Art.  341,  that  the  logarithm 
ol  any  root  of  a  number  is  equal  to  the  logarithm  of  that  num» 
5er  divided  by  the  index  of  the  root.     Hence,  to  extract  th< 
cot  of  a  number  by  logarithms,  we  havo  the  following 

B 


18  T  R  I  G  0  IN  0  M  E  T  R  Y. 

RULE. 

Duide  the  logarithm  of  the  number  by  the  index  oj  Int 
root  required. 

Ex.  1.  Required  the  cube  root  of  482.38. 

The  logarithm  of  482.38  is  2.683389. 

Dividing  by  3,  we  have  0.894463,  which  corresponds  to 
7.842,  which  is  therefore  the  root  required. 

Ex.  2.  Required  the  100th  root  of  365. 

Ans.,  1.0608. 

When  the  characteristic  of  the  logarithm  is  negative,  and  i* 
not  divisible  by  the  given  divisor,  we  may  increase  the  char- 
acteristic by  any  number  which  will  make  it  exactly  divisible, 
provided  we  prefix  an  equal  positive  number  to  the  decimal 
part  of  the  logarithm. 

Ex.  3.  Required  the  seventh  root  of  0.005846. 

The  logarithm  of  0.005846  is  3.766859,  which  may  be  writ- 
ten  7+4.766859. 

Dividing  by  7,  we  have  1.680980,  which  is  the  logarithm  oi 
4797,  which  is,  therefore,  the  root  required. 

This  result  may  be  verified  by  multiplying  1.680980  by  7, 
the  result  will  be  found  to  be  3.766860. 

Ex.  4.  Required  the  fifth  root  of  0.08452. 

Rx.  5.  Required  the  tenth  root  of  0.007815. 

PROPORTION  BY  LOGARITHMS. 

(16.)   The  fourth  term  of  a  proportion  is  found  by  multiply 
ing  together  the  second  and  third  terms,  and  dividing  by  the 
first.     Hence,  to  find  the  fourth  term  of  a  proportion  by  loga 
lithms, 

Add  the  logarithms  of  the  second  and  third  terms,  and  from 
their  sum  subtract  the  logarithm  of  the  first  term. 

Ex.  1.  Find  a  fourth  proportional  to  72.34,  2.519,  and  357.4S 

Ans.,  12.448. 

(17.)  When  one  logarithm  is  to  be  subtracted  from  anothej, 
it  may  be  more  convenient  to  convert  the  subtraction  into  au 
addition,  which  may  be  done  by  first  subtracting  the  given  leg. 
arithm  from  10,  adding  the  difference  to  the  other 
8 ml  aftorwnrd  rejecting  the  10. 


LOGARITHMS.  19 

The  difference  between  a  given  logarithm  and  10  is  called 
its  complement ;  and  this  is  easily  taken  from  the  table  by  be- 
ginning at  the  left  hand,  subtracting  each  figure  from  9,  ex- 
cept the  last  significant  figure  on  the  right,  which  must  be 
subtracted  from  10. 

To  subtract  one  logarithm  from  another  is  the  same  as  to 
fidd  its  complement,  and  then  reject  10  from  the  result  For 
it  —  b  is  equivalent  to  10— b  +  a— 10. 

To  work  a  proportion,  then,  by  logarithms,  we  must 
Add  the  complement  of  the  logarithm  of  the  first  term  f<* 
'he  logarithms  of  the  second  and  third  terms. 

The  characteristic  must  afterward  be  diminished  by  10. 
Ex.  2.  Find  a  fourth  proportional  to  6853,  489,  and  38750 
The  complement  of  the  logarithm  of  6853  is  6.164119 
The  logarithm  of  489  is  2.689309 

"  "  38750  is  4.588272 

The  fourth  term  is  2765,  whose  logarithm  is  3.441700. 
One  advantage  of  using  the  complement  of  the  first  term  in 
working  a  proportion  by  logarithms  is,  that  it  enables  us  to 
exhibit  the  operation  in  a  more  compact  form. 

Ex.  3.  Find  a  fourth  proportional  to  73.84,  658.3,  and  4872 

Ans. 
Ex.  4    Find  R  fcurth  proportional  to  5.745,  781.2,  and  512* 


BOOK   II. 


PLANE  TRIGONOMETRY, 

(18.)  TRIGONOMETRY  is  the  science  which  leaches  how  to  de- 
termine the  sevwt**  parts  of  a  triangle  from  having  certain 
parts  given. 

Plane  Trigonometry  treats  of  plane  triangles  ;  Spherical 
Trigonometry  treats  of  spherical  triangles. 

(19.)  The  circumference  of  every  circle  is  supposed  to  Le 
divided  into  360  equal  parts,  called  degrees  ;  each  degree  into 
60  minutes,  and  each  minute  into  60  seconds.  Degrees,  min- 
utes. and  seconds  are  designated  by  the  characters  °,  ',  '' 
Thus  23°  14'  35"  is  read  23  degrees,  14  minutes,  and  35  sec- 
onds. 

Since  an  angle  at  the  center  of  a  circle  is  measured  by  the 
arc  intercepted  by  its  sides,  a  right  angle  is  measured  by  90°, 
two  right  angles  by  180°,  and  four  right  angles  are  measured 
by  360°. 

(20.)  The  complement  of  an  arc  is  what  remains  after  sub 
tr  acting  the  arc  from  90°  •     Thus  the 
*rc   DF  is   the   complement  of  AF. 
The  complement  of  25°  15'  is  64°  45'. 

In  general,  if  we  represent  any  arc 
>y  A,  its  complement  is  90°—  A. 
Hence,  if  an  arc  is  greater  than  90°, 
its  complement  must  be  negative. 
Thus,  the  complement  of  100°  15'  is 
-10°  15'.  Since  the  two  acute  an- 
gles of  a  right-angled  triangle  are  to- 


E 


gether  equal  to  a  right  angle,  each  of  them  must  be  the 
plemont  of  the  other. 

(21.)  The  supplement  of  an  arc  is  vhai  remains  alter  sub- 
tracting the  arc  from  ISO3.  Thus  the  arc  BDF  is  the  supple* 
•nnnt  of  the  arc  AF.  The  supplement  of  25°  15'  is  154°  45'. 

In  general,  'f  we  represent  any  arc  by  A,  its  supplement  is 


PLANF    TRIGONOMETRY.  21 

ISO0  -  A  Hence,  if  an  arc  is  greater  than  180°,  its  supple- 
ment  must  be  negative.  Thus  the  supplement  of  200°  i^  —  20 !. 
Since  in  every  triangle  the  sum  of  the  three  angles  is  180°, 
either  angle  is  the  supplement  of  the  sum  of  the  other  two. 

(22.)  The  sine  of  an  arc  is  the  perpendicular  let  fall  from 
one  extremity  of  the  arc  on  the  radius  passing  through  the 
other  extremity.  Thus  FGr  is  the  sine  of  the  arc  AF,  or  of  th(: 
angle  ACF. 

Every  sine  is  half  the  chord  of  double  the  arc.  Thus  the 
sine  FG-  is  the  half  of  FH,  which  is  the  chord  of  the  arc  FAH, 
double  of  FA.  The  chord  which  subtends  the  sixth  part  of 
the  circumference,  or  the  chord  of  60°,  is  equal  to  the  radius 
(Geom.j  Prop.  IV.,  B.  VI.) ;  hence  the  sine  of  30°  is  equal  tn 
half  of  the  radius. 

(23.)  The  versed  sine  of  an  arc  is  that  part  of  the  diametei 
intercepted  between  the  sine  and  the  arc.  Thus  GrA  is  the. 
versed  sine  of  the  arc  AF. 

(24).  The  tangent  of  an  arc  is  the  line  which  touches  it  ai 
one  extremity,  and  is  terminated  by  a  line  drawn  from  the. 
cenlsr  through  the  other  extremity.  Thus  AI  is  the  tangent 
of  the  arc  AF,  or  of  the  angle  ACF. 

(25.)  The  secant  of  an  arc  is  the  line  drawn  from  the  cen- 
ter of  the  circle  through  one  extremity  of  the  arc,  and  is  Urn 
lied  by  the  tangent  drawn  through  the  other  extremity. 

Thus  CI  is  the  secant  of  the  arc  AF,  or  of  the  angle  ACF. 

(26.)  The  cosine  of  an  arc  is  the  sine  of  the  complement  of 
that  arc.  Thus  the  arc  DF,  being  the  complement  of  AF,  FK 
is  the  sine  of  the  arc  DF,  or  the  cosine  of  tho  arc  AF. 

The  cotangent  of  an  arc  is  the  tangent  of  the  complement 
of  that  arc.  Thus  DL  is  the  tangent  of  the  arc  DF,  or  the  co- 
tangent of  the  arc  AF. 

The  cosecant  of  an  arc  is  the  secant  of  the  complement  o( 
that  arc.  Thus  CL  is  the  secant  of  the  arc  DF,  or  the  coso- 
cant  of  the  arc  AF. 

In  genera],  if  we  repr  ent  any  angle  by  A, 
cos.      A=sine    (90°-A). 
cot.      A^tang.  (90° -A), 
cosec.  A=sec.     (90°  — A). 

Since,  in  a  right-angled  triangle,  either  of  the  acute  angle* 


•23 


TRIGONOMETRY. 


is  the  complement  of  the  other,  the  sine,  tangent,  and  secant 
<>f  cne  of  these  angles  is  the  cosine,  cotangent,  and  cosecan 
of  the  other. 

(27.)  The  sine,  tangent,  and  secant  of  an  arc  are  equal  te 
fcho  sine,  tangent,  and  secant  of  its  supplement.  Thus  FG  ii 
the  sine  of  the  arc  AF,  or  of  its  sup- 
plement, BDF.  Also,  AI,  the  tan- 
gent of  the  arc  AF,  is  equal  to  BM, 
the  tangent  of  the  arc  BDF.  And 
CI,  the  secant  of  the  arc  AF,  is  equal 
to  CM,  the  secant  of  the  arc  BDF. 

The  versed  sine  of  an  acute  angle, 
ACF,  is  equal  to  the  radius  minus 
the  cosine  CG.  The  versed  sine  of 
an  obtuse  angle,  BCF,  is  equal  to  ra- 
dius pins  the  cosine  CG ;  that  is,  to  BG. 

(28.)  The  relations  of  the  sine,  cosine,  &c.,  to  each  othei, 
may  be  derived  from  the  proportions  of  the  sides  of  similai 
triangles.  Thus  the  triangles  CGF,  CAI,  CDL,  being  similar, 
we  have, 

1.  CG  :  GF  :  :  CA  :  AI ;  that  is,  representing  the  arc  by  A, 
and  the  radius  of  the  circle  by  R,  cos.  A  :  sin.  A  :  :  R  :  tang.  A. 

R  sin.  A 

Whence  tang.  A  —  —     — — . 
cos.  A 

"2.  CG  :  :  CF  :  CA  :  CJ;  that  is,  cos.  A  :  R  :  :  R  :  sec.  A. 

Ra 

Whence  sec.  A= - 

cos.  A 

3  GF  :  CG  : :  CD  :  DL;  that  is,  sin,  A  :  cos.  A  :  :  R  :  cot.  A. 

_  R  cos.  A 

Whence  cot.  A= — : — r — •. 
sm.  A 

4  GF  :  CF  : :  CD  :  CL  ;  that  is,  sin.  A  :  R  : :  R  :  cosec.  A. 

T>3 

Whence  cosec.  A=— — r- 
sin.  A 

5.  AI :  AC  : :  CD  :  DL ;  that  is,  tang.  A  :  R  : :  R  :  cot.  A. 

Ra 

Whence  tang.  A.  =  — -— r-. 
cot,  A 

The  preceding  values  of  tangent  and  cotangent,  secant  and 
cosecant  will  be  frequently  referred  to  hereafter,  and  should 
bf  carefully  committed  to  memory. 


PLANE    T  R  i  c  o  N  j  M  E  T  R  v. 


Also,  in  the  right-angled  triangle  CGF,  we  find  CGr*-|-  3F'=- 
UF;  that  is,  sin.  2A+cos.2A=R2 ;  or, 

The  square  of  the  sine  of  an  arc,  together  with  the  square. 
tf  its  cosine^  is  equal  to  the  square  of  the  radius. 

Hence  sin.  A— ±  VR2  — cos.  2A. 
And  cos. 


,A=±  VRa— sin.  2A. 
(29  )  A  table  of  natural  sines  ^  tangents,  &c.,  is  a  table  giv- 
ing the  lengths  of  those  lines  for  different  angles  in  a  circle 
whose  radius  is  unity. 

Thus,  if  we  describe  a  circle  with  a  radius  of  one  inch,  and 
divide  the  circumference  into  equal  parts  of  ten  degrees,  we 
Khali  find 

the  sine  of  10°  equals  0.174  inch ; 


"       "       20°      "      0.342     "            P 
"       "       30°      "      0.500     " 
"       "       40°      "      0.643     " 
"       "       50°      "      0.766     " 
"       "       60°      "      0.866     " 
"       "       70°      "      0.940     " 
"       "       80°      "      0.985     " 
u       u       90°      "      1.000     " 

^ 

^ 

.K 

0  = 

\ 

\N 

w 

\ 

» 
31? 

y 

1 

If  \ve  draw  the  tangents  of  the  same  arcs,  we  shall  find 
the  tangent  of  10°  equals  0.176  inch ; 


tt 

20° 

"      0.364     " 

tt 

30° 

"      0.577     " 

tt 

40° 

"      0.839     " 

tt 

45° 

"      1.000     " 

tl 

50° 

"      1.192     " 

It 

60° 

"      1.732     " 

tt 

70° 

u      2.747     " 

It 

80° 

"      5.671     » 

tt 

90° 

"       infinite. 

Also,  if  we  draw  the  secanf  s  of  the  same 
wcs,  we  shall  find  that 

the  secant  of  10°  equals  1.015  inch; 
"  "         20°      "      1.064     " 

"  »         30°      "      1.155     " 

tt          «         40°      "      1.305     '« 


70" 


23  TRIGONOMETRY. 

the  secant  of  50°  equals  1.556  inch ; 
"         "         60°      "      2.000     " 
«         u         70°      "      2.924     " 
<•         "         80°      "      5.759     " 
"          "         90°      «      infinite. 

In  the  accompanying  table,  pages  116-133,  the  sines,  co- 
sines, tangents,  and  cotangents  are  given  for  every  minute  of 
the  quadrant  to  six  places  of  figures. 

(30.)  To  find  from  the  table  the  natural  sine,  cosine,  §c,} 
of  an  arc  or  angle. 

If  a  sine  is  required,  look  for  the  degrees  at  the  top  of  the 
page,  and  for  the  minutes  on  the  left ;  then,  directly  under  the 
given  number  of  degrees  at  the  top  of  the  page,  and  opposite 
to  the  minutes  on  the  left,  will  be  found  the  sine  required. 
Since  the  radius  of  the  circle  is  supposed  to  be  unity,  the  sims 
of  every  arc  below  90°  is  less  than  unity.  The  sines  are  ex- 
pressed in  decimal  parts  of  radius  ;  and,  although  the  decimal 
point  is  not  written  in  the  table,  it  must  always  be  prefixed 
As  the  first  two  figures  remain  the  same  for  a  great  many 
numbers  in  the  table,  they  are  only  inserted  for  every  ten  min- 
utes, and  the  vacant  places  must  be  supplied  from  the  two 
leading  figures  next  preceding  Thus,  on 

page  120,  the  sine  of  25°  11'  is  0.425516 ; 
page  126,    "        "       51°  34'  is  0.783332,  &c. 
The  tangents  are  found  in  a  similar  manner.     Thus 
the  tangent  of  31°  44'  is  0.618417  ; 

"  "         65°  27'  is  2.18923. 

The  same  number  in  the  table  is  both  the  sine  of  an  arc  arid 
the  cosine  of  its  complement.  The  degrees  for  the  cosines 
must  be  sought  at  the  bottom  of  the  page,  and  the  minutes  on 
the  right.  Thus, 

on  page  130,  the  cosine  of  16°  42'  is  0.957822  ; 
on  page  118,   "         "         73°  17'  is  0.287639,  &c. 
The  cotangents  are  found  in  the  same  manner.     TKus 
the  cotangent  of  19°  16'  is  2.86089 ; 
«  "          54°  53'  is  0.703246. 

It  is  not  necessary  to  extend  the  tables  beyond  a  qu?<?rant, 
oecause  the  sine  of  an  angle  is  equal  to  that  of  its  suppler?<>nl 
(Art.  27).  Thus 


PLANE    T RIG oyoy  ETR  r.       .  ^ 

the  sine  of  143°  24'  is  0.596225  ; 

••'  cosine        of  151°  23'  is  0.877844; 
"  tangent      of  132°  36'  is  1.08749  ; 
"    cotangent  of  116°    7'  is  0.490256,  &o. 
(31.)  If  a  sine  is  required  for  an  arc  consisting  of  degrees, 
minutes,  and  seconds,  we  must  make  an  allowance  for  the  sec- 
onds in  the  same  manner  as  was  directed  in  the  case  of  loga- 
rithms, Art.  7  ;  for,  within  certain  limits,  the  differences  ol  tho 
sines  are  proportional  to  the  differences  of  the  corresponding 
arcs.     Thus 

the  sine  of  34°  25'  is  .565207  ; 
"       ;<       34°  26'  is  .565447. 

The  difference  of  the  sines  corresponding  to  one  minute  ol 
arc,  or  60  seconds,  is  .000240.  The  proportional  part  for  1 '  is 
found  by  dividing  the  tabular  difference  by  60,  and  the  quo- 
tient, .000004,  is  placed  at  the  bottom  of  page  122,  in  the  col- 
umn headed  34°.  The  correction  for  any  number  of  seconds 
will  be  found  by  multiplying  the  proportional  part  for  1"  by 
the  number  of  seconds. 

Required  the  natural  sine  of  34°  25'  37". 
The  proportional  part  for  1",  being  multiplied  by  37,  becomes 
148,  which  is  the  correction  for  37".     Adding  this  to  the  sinn 
jf  34°  25',  we  find 

the  sine  of  34°  25'  37"  is  .565355. 

Since  the  proportional  part  for  1"  is  given  to  hundredths  of  a 
unit  in  the  sixth  place  of  figures,  after  we  have  multiplied  by 
the  given  number  of  seconds,  we  must  reject  the  last  two  fig- 
ures of  the  product. 

Tn  the  same  manaer  we  find 

the  cosine  of  56°  34'  28"  is  .550853. 

It  will  be  observed,  that  since  the  cosines  decrease  while 
the  arcs  increase,  the  correction  for  the  28"  is  to  be  subtracted 
from  the  cosine  of  56°  34'. 
In  the  same  manner  we  find 

the  natural  sine  of  27°  17'  12"  is  0.458443  ; 

"         "      cosine        of  45°  23'  23"  is  0.702281 ; 
•«         "      tangent      of  63'  32'  34"  is  2.00945  : 
"         «      cotangent  of  81°  48'  56"  is  0  143825 


2<>  TRIGONOMETRY. 

(32.)  To  find  the  rur/iber  of  degrees,  minutes ,  and  second* 
belonging  to  a  given  sine  or  tangent 

If  the  given  sine  or  tangent  is  found  exactly  in  the  table, 
the  corresponding  degrees  will  be  found  at  the  top  of  the  page, 
and  the  minutes  on  the  left  hand.  But  when  the  given  num- 
ber is  not  found  exactly  in  the  table,  look  for  the  sine  or  tan- 
gont  which  is  next  less  than  the  proposed  one,  and  take  out 
the  corresponding  degrees  and  minutes.  Find,  also,  the  dif- 
ference between  this  tabular  number  and  the  number  proposed, 
and  divide  it  by  the  proportional  part  for  1"  found  at  the  hot 
f.om  of  the  page  ;  the  quotient  will  be  the  required  number  of 
seconds. 

Required  the  arc  whose  sine  is  .750000. 

The  next  less  sine  in  the  table  is  .749919,  the  arc  correspond- 
ing to  which  is  48°  35'.  The  difference  between  this  sine  and 
that  proposed  is  81,  which,  divided  by  3.21,  gives  25.  Hencr 
the  required  arc  is  48°  35'  25". 

Tn  the  same  manner  we  find 

the  arc  whose  tangent  is  2.00000  is  63°  26'  6". 

If  a  cosine  or  cotangent  is  required,  we  must  look  for  the 
number  in  the  table  which  is  next  greater  than  the  one  pro 
posed,  and  then  proceed  as  for  a  sine  or  tangent.     Thus 
the  arc  whose  cosine        is    .40000  is  66°  25'  18"  ; 
"     "        "      cotangent  is  1.99468  is  26°  37'  34". 

(33.)  On  pages  134-5  will  be  found  a  table  of  natural  se- 
oants  for  every  ten  minutes  of  the  quadrant,  carried  to  seven 
places  of  figures.  The  degrees  are  arranged  in  order  in  tho 
first  vertical  column  on  the  left,  and  the  minutes  at  the  top 
nf  the  page.  Thus 

the  secant  of  21°  20'  is  1.073561  ; 
«         "         81°  50'  is  7.039622. 

if  a  secant  is  required  for  a  number  of  minutes  not  given  in 
the  table,  the  correction  for  the  odd  minutes  may  be  found  by 
means  of  tho  last  vertical  column  on  the  right,  which  shows 
the  proportional  part  for  one  minute. 

Let  it  be  required  to  find  the  secant  of  30°  33' 
The  secant  of  30°  30'  is  1.160592. 

The  correction  for  1'  is  198.9,  which,  multiplied  by  3.  bo- 


PLANE    TR  IGONOMETIIT.  27 

comes  597.     Adding  this  to  the  number  before  found,  we  ob. 
tain  1.161189. 

For  a  cosecant,  the  degrees  must  be  sought  in  the  right- 
hand  vertical  column,  and  the  minutes  at  the  bottom  of  th« 
page.  Thus 

the  cosecant  of  47°  40'  is  1.352742  ; 
«  «         38°  33'  is  1.604626. 

(34.)  When  the  natural  sines,  tangents,  &c.,  are  used  in  pro 
portions,  it  is  necessary  to  perform  the  tedious  operations  of 
multiplication  and  division.  It  is,  therefore,  generally  prefer- 
able to  employ  the  logarithms  of  the  sines ;  and,  for  conven- 
ience, these  numbers  are  arranged  in  a  separate  table,  called 
logarithmic  sines,  &c.  Thus 

the  natural  sine  of  14°  30'  is  0.250380. 

Its  logarithm,  found  from  page  6,  is  1.398600. 

The  characteristic  of  the  logarithm  is  negative,  as  must  be 
the  case  with  all  the  sines,  since  they  are  less  than  unity.  Te 
avoid  the  introduction  of  negative  numbers  in  the  table,  we  in- 
crease the  characteristic  by  10.  making  9.398600,  and  this  is 
the  number  found  on  page  38  for  the  logarithmic  sine  of  14° 
30'.  The  radius  of  the  table  of  logarithmic  sines  is  therefore, 
properly,  10,000,000,000,  whose  logarithm  is  10. 

(35.)  The  accompanying  table  contains  the  logarithmic  sinea 
and  tangents  for  every  ten  seconds  of  the  quadrant.  The  de- 
grees and  seconds  are  placed  at  the  top  of  the  page,  and  the 
minutes  in  the  left  vertical  column.  After  the  first  two  de- 
grees, the  three  leading  figures  in  the  table  of  sines  are  only 
given  in  the  column  headed  0",  and  are  to  be  prefixed  to  the 
numbers  in  the  other  columns,  as  in  the  table  of  logarithms  of 
numbers.  Also,  where  the  leading  figures  change,  this  change 
is  indicated  by  dots,  as  in  the  former  table.  The  correction 
for  any  number  of  seconds  less  than  10  is  given  at  the  bottom 
uf  the  page. 

(36.)  To  find  the  logarithmic  sine  or  tangent  of  a  given 
arc. 

Look  for  the  degrees  at  the  top  of  the  page,  the  minutss  on 
the  left  hand,  and  the  next  less  number  of  seconds  at  the  tor>; 
then,  under  the  seconds,  and  opposite  to  the  minutes,  will  be 
tound  four  figures,  to  which  the  three  leading  figures  are  to  b« 


2%  TlUGONOMETR*. 

prefixed  from  tht  column  headed  Ox/ ;  to  this  add  the  proper 
tional  part  for  the  odd  seconds  at  the  bottom  of  the  page. 
Required  the  logarithmic  sine  of  24°  27'  34". 

The  logarithmic  sine  of  24°  27'  30"  is  9.617033 
Proportional  part  for  4"  is  18 

Logarithmic  sine  of  24°  27'  34"  is       9.617051. 
Required  the  logarithmic  tangent  of  73°  35'  43". 
The  logarithmic  tangent  73°  35'  40"  is  10.531031 
Proportional  part  for  3"  is  23 

Logarithmic  tangent  of  73°  35'  43"  is    10.531054. 
When  a  cosine  is  required,  the  degrees  and  seconds  must  be 
sought  at  the  bottom  of  the  page,  and  the  minutes  on  the  right, 
and  the  correction  for  the  odd  seconds  must  be  subtracted  from 
the  number  in  the  table. 

Required  the  logarithmic  cosine  of  59°  33'  47". 

The  logarithmic  cosine  of  59°  33'  40"  is  9.704682 
Proportional  part  for  7"  is  25 

Logarithmic  cosine  of  59°  33'  47"  is        9.704657. 
So,  also,  the  logarithmic  cotangent  of  37°  27'  14"  is  found 
to  be  10.115744. 

It  will  be  observed  that  for  the  cosines  and  cotangents,  the 
seconds  are  numbered  from  10"  to  60",  so  that  if  it  is  re* 
quired  to  find  the  cosine  of  25°  25'  0"  we  must  look  for  25C 
24'  60" ;  and  so,  also,  for  the  cotangents. 

(37.)  The  proportional  parts  given  at  the  bottom  of  each 
page  correspond  to  the  degrees  at  the  top  of  the  page,  in- 
creased by  30',  and  are  not  strictly  applicable  to  any  other 
number  of  minutes ;  nevertheless,  the  differences  of  the  sines 
change  so  slowly,  except  near  the  commencement  of  the  quad- 
rant, that  the  error  resulting  from  using  these  numbers  for 
every  part  of  the  page  will  seldom  exceed  a  unit  in  the  sixth 
decimal  place.  For  the  first  two  degrees,  the  differences 
change  so  rapidly  that  the  proportional  part  for  1"  is  given  for 
*ach  minute  in  the  right-hand  column  of  the  page.  The  cor- 
rcction  for  any  number  of  seconds  less  than  ten  \\ill  be  foum 
by  multiplying  the  proportional  part  for  1"  by  the  given  nun> 
ber  of  seconds. 

Required  the  logarithmic  sine  of  1°  17'  Go". 


PLANS    TRIGONOME  TRY.  2& 

The  Lgarithmic  sine  of  1°  17'  30"  is  8.352991. 

The  correction  for  o"  is  found  by  multiplying  93.4  by  3, 
which  gives  280.  Adding  this  to  the  above  tabular  number, 
we  obtain  for 

the  sine  of  1°  17'  33",  8.353271. 

A  similar  method  may  be  employed  for  several  of  the  first 
degrees  of  the  quadrant,  if  the  proportional  parts  at  the  bottom 
of  the  page  are  not  thought  sufficiently  precise.  This  correc 
tioa  may,  however,  be  obtained  pretty  nearly  by  inspection, 
from  comparing  the  proportional  parts  for  two  successive  de- 
grees. Thus,  on  page  26,  the  correction  for  1",  corresponding 
to  the  sine  of  2°  30',  is  48 ;  the  correction  for  1",  correspond- 
ing to  the  sine  of  3°  30',  is  34.  Hence  the  correction  for  1", 
corresponding  to  the  sine  of  3°  0',  must  be  about  41 ;  and,  in 
the  same  manner,  we  may  proceed  for  any  other  part  of  the 
table. 

(38.)  .Near  the  close  of  the  quadrant,  the  tangents  vary  su 
rapidly  that  the  same  arrangement  of  the  table  is  adopted  as 
for  the  commencement  of  the  quadrant.  For  the  last,  as  well 
as  the  first  two  degrees  of  the  quadrant,  the  proportional  part 
to  1"  is  given  for  each  minute  separately.  These  proportional 
parts  are  computed  for  the  minutes  placed  opposite  to  them, 
increased  by  30",  and  are  not  strictly  applicable  to  any  other 
number  of  seconds ;  nevertheless,  the  differences  for  the  most 
part  change  so  slowly,  that  the  error  resulting  from  using  these 
numbers  for  every  part  of  the  same  horizontal  line  is  quite 
small.  When  great  accuracy  is  required,  the  table  on  page  114 
may  be  employed  for  arcs  near  the  limits  of  the  quadrant.  This 
table  furnishes  the  differences  between  the  logarithmic  sinca 
and  the  logarithms  of  the  arcs  expressed  in  seconds.  Thus 

the  logarithmic  sine  of  0°  5',  from  page  22,  is  7.162696 

the  logarithm  of  300"  (=-5')  is  2.477121 

the  difference  is  4.685575. 

This  is  the  number  found  on  page  114,  under  the  heading 
log.  sine  A— log.  A",  opposite  to  5  min. ;  and,  in  a  similar  man- 
ner, the  other  numbers  in  the  same  column  are  obtained.  These 
numbers  vary  quite  slowly  for  two  degrees  ;  and  henc°,  to  find 
I  he  logarithmic  sine  of  an  arc  less  than  two  degrees  we 


30  T  R  I  G  0  N  0  M  E  T  R  7. 

but  to  add  ths  logarithm  of  the  arc  expressed  in  seconds  to  thf 

appropriate  number  found  in  this  table. 
Required  the  logarithmic  sine  of  0°  7'  22 v. 

Tabular  number  from  page  114,  4.685575 
The  logarithm  of  442"  is  2.645422 

Logarithmic  sine  of  0°  T  22"  is  7.330997. 
The  logarithmic  tangent  of  an  arc  less  than  two  degrees  la 
found  in  a  similar  manner. 
Required  the  logarithmic  tangent  of  0°  27'  36". 

Tabular  number  from  page  114,  4.685584 

The  logarithm  of  1656"  is  3.219060 

Logarithmic  tangent  of  0°  27'  36"  is  7.904644 
The  column  headed  log.  cot.  A+log.  A",  is  found  by  adding 
f.hc  logarithmic  cotangent  to  the  logarithm  of  the  arc  expressed 
in  seconds.  Hence,  to  find  the  logarithmic  cotangent  of  an  arc 
less  than  two  degrees,  we  must  subtract  from  the  tabular  num- 
ber the  logarithm  of  the  arc  in  seconds. 

Required  the  logarithmic  cotangent  of  0°  27'  36". 
Tabular  number  from  page  114,  15.314416 

The  logarithm  of  1656"  is  3.219060 

Logarithmic  cotangent  of  0D  27'  36"  is  12.095356. 

The  same  method  will,  of  course .  furnish  cosines  and  cotan 
gents  of  arcs  near  90°. 

(39.)  The  secants  and  cosecants  are  omitted  in  this  table, 
since  they  are  easily  derived  from  tho  cosines  and  sines.  Wo 

R2 

have  found,  Art.  28,  secant  = — : —  ;  or,  taking  the  logarithms, 

cosine ' 

log.  secant  =2.  log.  R  — log.  cosine 

—20— log.  cosine. 

R2 

Also,  cosecant          =  - — , 

sine 

or  log.  cosecant        =20— log.  sine.     That  is, 

The  logarithmic  secant  is  found  by  subtracting  the  logo** 
rilhmic  cosine  from  20;  and  the  logarithmic  cosecant  is  found 
by  subtracting  the  logarithmic  sine  from  20. 

Thus  we  have  found  tho  logarithmic  sine  of  24°  27'  34'  tc 
be  9.617051. 

Hence  the  logarithmic  c  .secant  of  24°  27'  34"  is  10.382949 


PLANE    TRIGONOMETRY,  3\ 

The  logarithmic  cosine  of  54°  12'  40"  is    9.76700S 

Hence  the  logarithmic  secant  of  54°  12'  40"  is  10.232992 

(40.)  To  find  the  arc  corresponding-  to  a  given  logarithms, 
line  or  tangent. 

If  the  given  number  is  found  exactly  in  the  table,  the  cor- 
responding degrees  and  seconds  will  be  found  at  the  top  of  the 
page,  and  the  minutes  on  the  left.  But  when  the  given  num- 
ber is  not  found  exactly  in  the  table,  look  for  the  sine  or  tan- 
gent which  is  next  less  than  the  proposed  one,  and  take  out 
the  corresponding  degrees,  minutes,  and  seconds.  Find,  also, 
the  difference  between  this  tabular  number  and  the  number 
proposed,  and  corresponding  to  this  difference,  at  the  bottom 
of  the  page,  will  be  found  a  certain  number  of  seconds  which 
is  to  be  added  to  the  arc  before  found. 

Required  the  arc  corresponding  to  the  logarithmic  sine 
9.750000. 

The  next  less  sine  in  the  table  is  9.749987. 

The  arc  corresponding  to  which  is  34°  13'  0". 

The  difference  between  its  sine  and  the  one  proposed  is  13, 
corresponding  to  which,  at  the  bottom  of  the  page,  we  find  4" 
nearly.  Hence  the  required  arc  is  34°  13'  4". 

In  the  same  manner,  we  find  the  arc  corresponding  to  loga- 
rithmic tangent  10.250000  to  be  60°  38'  57". 

When  the  arc  falls  within  the  first  two  degrees  of  the  quad- 
rant, the  odd  seconds  may  be  found  by  dividing  the  difference 
between  the  tabular  number  and  the  one  proposed,  by  the  pro- 
portional part  for  1".  We  thus  find  the  arc  corresponding  to 
logarithmic  sine  8.400000  to  be  1°  26'  22"  nearly. 

We  may  employ  the  same  method  for  the  last  two  degrees 
of  the  quadrant  when  a  tangent  is  given  ;  but  near  the  limits 
of  the  quadrant  it  is  better  to  employ  the  auxiliary  table  on 
page  114.  The  tabular  number  on  page  114  is  equal  to  log. 
sin.  A— log.  A".  Hence  log.  sin.  A— tabular  number  =log, 
A"  ;  that  is,  if  we  subtract  the  corresponding  tabular  number 
on  page  114,  from  the  given  logarithmic  sine,  the  remainder 
will  be  the  logarithm  of  the  arc  expressed  in  seconds. 

Required  the  arc  corresponding  to  logarithmic  sine  7.000000. 

We  see,  from  page  22,  that  the  arc  must  be  nearly  3' ;  tn.0 
•(>or  respond  ing  tabular  number  on  page  114  is  4.685575 


32  T  K  I  G  O  N  O  M.  K  T  R  V. 

The  difference  is  2.314425, 
which  is  the  logarithm  of  206."265. 

Hence  tho  required  arc  is  3'  26. "265. 

Required  the  arc  corresponding  to  log.  sine  8.000000. 

We  see  from  page  22,  that  the  arc  is  ahout  34'.  The  co/« 
responding  tabular  number  from  page  114  is  4.685563,  which; 
subtracted  from  8.000000,  leaves  3.314432,  which  is  the  log- 
arithm  of  2062. "68.  Hence  the  required  arc  is 

34'  22."68. 

In  the  same  manner,  we  find  the  arc  corresponding  to  loga 
nthmio  tangent  8.184608  to  be  0°  52'  35". 

SOLUTIONS  OF  RIGHT-ANGLED  TRIANGLES. 

THEOREM  I. 

(41.)  In  any  right-angled  triangle,  radius  is  to  the  hypoth* 
enuse  as  the  sine  of  either  acute  angle  is  to  the  opposite.  side, 
or  the  cosine  of  either  acute  angle  to  the  adjacent  side. 

Let  the  triangle  CAB  be  right  angled 
at  A,  then  will 

R  :  CB  :  :  sin.  C  :  BA  : :  cos.  C  :  CA. 
From  the  point  C  as  a  center,  with  a 


radius  equal  to  the  radius  of  the  tables,   C 
describe  the  arc  DE,  and  on  AC  let  fall  the  perpendicular  EF 
Then  EF  will  be  the  sine,  and  CF  the  cosine  of  the  angle  C 
Because  the  triangles  CAB,  CFE  are  similar,  we  have 

CE  :  CB  :  :  EF  :  BA, 

or  E,  :  CB  :  :  sin.  C  :  BA. 

Als.1,  CE  :  CB  :  :  CF  :  CA, 

r.r  R  :  CB  :  :  cos.  C  :  CA. 

THEOREM  II. 

(42.)  In  any  right-angled  triangle,  radius  is  to  either  sidt 
as  the  tangent  of  the  adjacent  acute  angle  is  to  the  opposite 
side,  or  the  secant  of  the  same  angle  to  the  hypothenuse. 

Let  the  triangle  CAB  be  right  angled 
at  A,  then  will 

R  :  CA  : :  tang.  C  :  AB  : :  sec.  C  :  CB. 

From  the  point  C  as  a  center,  with  a 
radius  equal  to  tho  radius  of  the  tables,    c 


PLANE    TRIGONOMETRY.  H3 

describe  the  arc  BE,  and  from  the  point  D  draw  DF  perpen- 
dicular to  CA.     Then  DF  will  "be  the  tangent,  and  CF  the  se- 
cant of  the  angle  C.     Because  the  triangles  CAB,  CDF  aro 
similar,  we  have    CD  :  CA  :  :  DF  :  AB, 
or  R  :  CA  :  :  tang.  C  :  AB. 

Also,  CD  :  CA  :  :  CF  :  CB, 

or  R  :  CA  :  :  sec.  C  :  CB. 

(43.)  In  every  plane  triangle  there  are  six  parts  :  three  sides 
and  three  angles.  Of  these,  any  three  being  given,  provided 
one  of  them  is  a  side,  the  others  may  be  determined.  In  a 
right-angled  triangle,  one  of  the  six  parts,  viz.,  the  right  anglo, 
is  always  given ;  and  if  one  of  the  acute  angles  is  given,  the 
other  is,  of  course,  known.  Hence  the  number  of  parts  to  be 
considered  in  a  right-angled  triangle  is  reduced  to  four,  any 
two  of  which  being  given,  the  others  may  be  found. 

It  is  desirable  to  .have  appropriate  names  by  which  to  des 
ignate  each  of  the  parts  of  a  triangle.  One  of  the  sides  ad- 
jacent to  the  right  angle  being  called  the  base,  the  other  side 
adjacent  to  the  right  angle  may  be  called  the  perpendicular. 
The  three  sides  will  then  be  called  the  hypothenuse,  base,  and 
perpendicular.  The  base  and  perpendicular  are  sometimes 
called  the  legs  of  the  triangle.  Of  the  two  acute  angles,  that 
which  is  adjacent  to  the  base  may  be  called  the  angle  at  the 
base,  and  the  other  the  angle  at  the  perpendicular. 

We  may,  therefore,  have  four  cases,  according  as  there  are 
given, 

1.  The  hypothenuse  and  the  angles  : 

2.  The  hypothenuse  and  a  leg ; 

3.  One  leg  and  the  angles  ;  or, 

4.  The  two  legs. 

All  of  these  cases  may  be  solved  by  the  tfro  preceding  iheo 
rcms. 

CASE  I. 

(44.)  Given  the  hypothenuse  and  the  angles,  to  find  the  base 
and  perpendicular. 

This  case  is  solved  by  Theorem  I. 

Radius  :  hypothenuse  : :  sine  of  the  angle  at  the  base  :  per- 
pendicular ; 

: :  cosine  of  the  angle  at  ihe  base  :  base 
C 


34  TRIGONOMETRY. 

Ex.  1,  Given  the  hypothenuae  275,  and  the  angle  at  the  bas« 
57°  23',  to  find  the  base  and  perpendicular. 
The  natural  sine  of  57°  23'  is  .842296 ; 
"  cosine        "  .539016. 

Hence  1  :  275  :  :  .842296  :  231.631=AB. 
1  :  275  :  :  .539016  :  148.229=AC. 
The  computation  is  here  made  by  natural 
numbers.     If  we  work  the  proportion  by  loga- 
rithms, we  shall  have 

Radius,  10.000000 

Is  to  the  hypothenuse  275  2.439333 

As  the  sine  of  C  57°  23'  9.925465 

To  the  perpendicular  231.63  2.364798. 

Also,  Radius,  10.000000 

Is  to  the  hypothenuse  275  2.439333 

As  the  cosine  of  C  57°  23'  9.731602 

To  the  base  148.23  2.170935, 

Ex.  2.  Given  the  hypothenuse  67.43,  and  the  angle  at  tin* 
perpendicular  38°  43',  to  find  the  base  and  perpendicular. 

Ans.  The  base  is  42.175,  and  perpendicular  52.612. 

The  student  should  work  this  and  the  following  examples 

both  by  natural  numbers  and  by  logarithms,  until  he  has  made 

himself  perfectly  familiar  with  both  methods.     He  may  then 

p.mploy  either  method,  as  may  appear  to  him  most  expeditious. 

CASE  II. 

(45.)  Given  the  hypothenuse  and  one  leg-,  to  find  the  angles 
and  the  other  leg-. 

This  case  is  solVed  by  Theorem  I. 

Hypothenuse  :  radius  : :  base  :  cosine  of  the  ang-le  at  the  base 
Radius  :  hypothenuse  : :  sine  of  the  angle  at  the  base  : 
perpendicular. 

When  the  perpendicular  is  given,  perpendicular  must  b« 
substituted  for  base  in  this  proportion. 

Ex  1.  Given  the  hypothenuse  54.32,  and  the  ba<?e  32.11,  t<: 
find  the  angles  and  the  perpendicular. 

By  natural  numbers,  we  have 


PLANF    TRIGONOMETRY.  3f» 

54.32  :  1  :  :  32.11  :  .591127,  which  is  the  cosine  of  53°  45 
17'  ,  the  angl  5  at  the  base. 

Also,  1  :  54  32  :  :  .806580  :  43.813=the  perpendicular. 

The  computation  may  be  performed  more  expeditiously  ty 
logarithms,  as  in  the  former  case. 

Ex.  2.  G-iven  the  hypothenuse  332.49,  and  the  perpendicu- 
[ar  98.399,  to  find  the  angles  and  the  base. 

Ans.  The  angles  are  17°  12'  51"  and  72°  47'  9" ;  the  base, 
317.6. 

CASE  III. 

(46.)  Given  one  leg"  and  the  angles,  to  find  the  other  leg 
and  hypothenuse. 

This  case  is  solved  by  Theorem  II. 

Radius  :  base  : :  tangent  of  the  angle  at  the  base  :  the  perpen 
dicular. 

: :  secant  of  the  angle  at  the  base  :  hypothenuse. 

When  the  perpendicular  is  given,  perpendicular  must  be 
substituted  for  base  in  this  proportion. 

Ex.  1.  Given  the  base  222,  and  the  angle  at  the  base  25°  15', 
to  find  the  perpendicular  and  hypothenuse. 

By  natural  numbers,  we  have 

1  :  222  :  :    .471631  :  104.70,  perpendicular ; 
:  :  1.105638  :  245.45,  hypothenuse. 

The  computation  should  also  be  performed  by  logarithms,  as 
in  Case  I. 

Ex.  2.  G-iven  the  perpendicular  125,  and  the  angle  at  the 
perpendicular  51C  19',  to  find  the  hypothenuse  and  base. 

Ans.  Hypothenuse,  199.99 ;  base,  156.12 

CASE  IY. 

(47.)  Given  the  two  legs,  to  find  the  angles  and  hypothenuse. 

This  case  is  solved  by  Theorem  II. 

Base  :  radius  :  -.perpendicular :  tangent  of  the  angle  at  the  base. 
Radius  :  base : :  secant  of  the  angle  at  the  base  :  hypothenuse. 

Ex.  1.  Given  the  base  123,  and  perpendicular  765,  to  find 
the  angles  and  hypothenuse. 

By  natural  numbers,  we  have 

123  :  1  : :  765  :  6.219512,  which  is  the  tangent  of  80°  51 
57",  the  angle  at  the  base. 


36  T  R  I  G  f )  N  O  M  E  T  R  Y. 

1.  :  123  :  :  6.299338  :  774.82,  hypothenuse. 

The  computation  may  also  be  made  oy  logai  ithms,  as  in 
Case  I. 

Ex.  2.  Given  the  base  53,  and  perpendicular  67,  to  find  the 
angles  and  hypothenuse. 

Ans.  The  angles  are  51°  39'  16"  and  38°  20'  44"  ;  hypothe- 
nuse,  85.428. 

Examples  for  Practice. 

1.  Given  the  base  777,  and  perpendicular  345,  to  find  the 
hypothenuse  and  angles. 

This  example,  it  will  be  seen,  falls  under  Case  IV. 

2.  Given  the  hypothenuse  324,  and  the  angle  at  the  base 
48°  17',  to  find  the  base  and  perpendicular. 

3.  Given  the  perpendicular  543,  and  the  angle  at  the  basu 
72°  45',  to  find  the  hypothenuse  and  base. 

4.  Given  the  hypothenuse  666,  and  base  432,  to  find  the  an- 
gles and  perpendicular. 

5.  Given  the  base  634,  and  the  angle  at  the  base  53°  27',  tu 
find  the  hypothenuse  and  perpendicular. 

6.  Given  the  hypothenuse  1234,  and  perpendicular  555,  to 
find  the  base  and  angles. 

(48.)  "When  two  sides  of  a  right-angled  triangle  are  given, 
the  third  may  be  found  by  means  of  the  property  that  the 
square  of  the  hypothenuse  is  equal  to  the  sum  of  the  squares 
of  the  other  two  sides. 

Hence,  representing  the  hypothenuse,  base,  and  perpendicu 
lar  by  the  initial  letters  of  these  words,  we  have 

Ex.  1.  If  the  base  is  2720,  and  the  perpendicular  3104,  what 
is  the  hypothenuse  ?  Ans.,  4127.1 

Ex.  2.  If  the  hypothenuse  is  514,  and  the  perpendicular  132, 
what  is  the  base  ? 

SOLUTIONS  OP'  OBLIQUE-ANGLED  TRIANGLES. 

THEOREM  I. 

(49.)  In  any  plane  triangle,  the  sines  of  the  angles  art 
proportional  to  the  opposite  sides 


PLANI:    TRIG  ON  D  MM  TRY. 


87 


D 


Let  ABC  be  any  triangle,  and  from  one 
of  its  angles,  as  C,  let  CD  be  drawn  per- 
pendicular to  AB.  Then,  because  the 
triangle  ACD  is  right  angled  at  D,  we 
have 

R  :  sin.  A  :  :  AC  :  CD  ;  whence  RxC.D=sin.  AxAC. 

For  the  same  reason, 
R  :  sin.  B  :  :  BC  :  CD ;  whence  RxC.D=sm.  BxBC. 

Therefore,          sin.  AxAC=sin.  BxBC, 
or  sin.  A  :  sin.  B  :  :  BC  :  AC. 

THEOREM  II. 

(50.)  In  any  plane  triangle,  the  sum  of  any  two  sides  is  la 
their  difference,  as  the  tangent  of  half  the  sum  of  the  opposite 
angles  is  to  the  tangent  of  half  their  difference. 

Let  ABC  be  any  triangle ;  then  will 


CB-hCA  :  CB-CA 


tang. 


A+B 


tang.  - 


A-B 


,D 


2  2 

Produce  AC  to  D,  making  CD  equal  to  CB,  and  join  DD. 
I1ake  CE  equal  to  CA,  draw  AE,  and  produce  it  to  F.  Then 
AD  is  the  sum  of  CB  and  CA,  and  BE  is  their  difference. 

The  sum  of  the  two  angles  CAE,  CEA,  is  equal  to  the  sura 
of  CAB,  CBA,  each  being  the  supplement  of  ACB  (Geom.. 
Prop.  27,  B.  I.).  But,  rince  CA  is  equal 
to  CE,  the  angle  CAE  is  equal  to  the  an- 
gle CEA;  therefore,  CAE  is  the  half 
sum  of  the  angles  CAB,  CBA.  Also,  if 
from  the  greater  of  the  two  angles  CAB, 
CBA,  there  be  taken  their  half  sum,  the 
remainder,  FAB,  will  be  their  half  differ- 
ence (Algebra,  p.  68). 

Since  CD  is  equal  to  CB,  the  angle  ADF 
is  equal  to  the  angle  EBF ;  also,  the  an- 
gle CAE  is  equal  to  AEC,  which  is  equal 
to  the  vertical  angle  BEF.  Therefore,  the  two  triangles  DAP, 
BEF,  are  mutually  equiangular ;  hence  the  two  angles  at  P 
are  equal,  and  AF  is  perpendicular  to  DB.  If,  then,  AF  bn 
made  radius,  DF  will  be  the  tangent  of  DAF,  and  BF  will  hti 
the  tangent  of  BAF.  But,  by  similar  triangles,  we  have 


38  T  R  I  b  O  >'  0  M  E  T  R  Y.  . 

AD  :  BE  :  :  DF  :  BF  ;  that  is, 
OB-f-CA  :  CB-CA  :  :  tang.  -~  :  tang. 


THEOREM  III. 

(51.)  If  from  any  angle  of  a  triangle  a  perpendicular  ot 
drawn  to  the  opposite  side  or  base,  the  whole  base  will  be  'tc 
the  sum  of  the  other  two  sides,  as  the  difference  of  those  twn 
sides  is  to  the  difference  of  the  segments  of  the  base. 

For  demonstration,  see  Geometry,  Prop.  31,  Cor.,  B.  IV. 

(52).  In  every  plane  triangle,  three  parts  must  be  given  ic 
enable  us  to  determine  the  others  ;  and  of  the  given  parts,  one> 
at  least,  must  be  a  side.  For  if  the  angles  only  are  given, 
i  hese  might  belong  to  an  infinite  number  of  different  triangles 
in  solving  oblique-angled  triangles,  four  different  ca^s  may 
therefore  be  presented.  There  may  be  given, 

1.  Two  angles  and  a  side  ; 

2.  Two  sides  and  an  angle  opposite  one  of  them  ; 

3.  Two  sides  and  the  included  angle  ;  or, 

4.  The  three  sides. 

We  shall  represent  the  three  angles  of  the  proposed  triangle 
by  A,  B,  C,  and  the  sides  opposite  them,  respectively,  by  a,  b,  c 

CASE  I. 

(53.)  Given  two  angles  and  a  side,  to  find  the  tliiid  angle 
ind  the  other  two  sides. 

To  find  the  third  angle,  add  the  given  angles  together,  and 
subtract  their  sum  from  180°. 

The  required  sides  may  be  found  by  Theorem  I.  The  pro- 
jjortion  will  be, 

The  sine  of  the  angle  opposite  the  given  side  :  the  given  side 

:  :  the  sine  of  the  angle  opposite  the  required  side  :  the  re- 
quired side. 

Ex.  1.  In  the  triangle  ABC,  there  are 
given  the  angle  A,  57°  15',  the  angle  B, 
35°  30',  and  the  side  c,  364,  to  find  the 
«ther  parts. 

The  sum  of  the  given.  angles,  subtracted  A 


PLANE    TRIGONOMETRY  3y 

180°,  leaves  87°  15'  for  the  angle  C.     Then,  to  find  the 
side  <z,  we  say,         sin.  C  :  c  : :  sin.  A  :  a. 
By  natural  numbers, 

.998848  :  364  : :  .841039  :  306.49 =a. 
This  proportion  is  most  easily  worked  by  logarithms,  thus , 
As  the  sine  of  the  angle  C,  87°  15',  comp.,  0.000500 
Is  to  the  side  c,  364,  2.561101 

So  is  the  sine  of  the  angle  A,  57°  15',       9.924816 
To  the  side  a,  306.49,  2.486417 

To  find  the  side  b  : 

sin.  C  ;  c  :  :  sin.  B  :  b. 
By  natural  numbers, 

.998848  :  364  : :  .580703  :  211.62=£. 
The  work  by  logarithms  is  as  follows  : 

sin.  C,  87°  15',  comp.,  0.000500 

:  c,  364,  2.561101 

: :  sin.  B,  35°  30',  9.763954 

:  b,  211.62,  2.325555. 

Ex.  2.  In  the  triangle  ABC,  there  are  given  the  angle  AT 
49°  25',  the  angle  C,  63°  48',  and  the  side  c,  275,  to  find  the 
other  Darts.  Ans.,  B=66°  47' ;  0=232.766  ;  £=281.67. 

CASE  II. 

(54.)  Given  two  sides  and  an  angle  opposite  one  of  them, 
to  find  the  third  side  and  the  remaining1  angles. 

One  of  the  required  angles  is  found  by  Theorem  I.  Tho 
proportion  is, 

The  s'<  de  opposite  the  given  angle  :  the  sine  of  that  angle 

:  :  the  other  given  side  :  the  sine  of  the  opposite  angle. 

The  third  angle  is  found  by  subtracting  the  sum  of  the  other 
tvro  from  180°  ;  and  the  third  side  is  found  as  in  Case  I. 

If  the  side  BC,  opposite  the  given  an-  « 

gle  A,  is  shorter  than  the  other  given  side 
\C,  the  solution  will  be  ambiguous  ;  that 

*s,  two  different  triangles,  ABC,  AB'C,     /  ..  / *, 

may  be  formed,  each  of  which  will  satisfy  A      B"- •"*  B' 

the  conditions  of  the  problem. 

The  numerical  result  is  also  ambiguous,  for  the  fourth  term 


40  TRIGONOMETRY 

of  the  first  proportion  is  a  sine  of  an  angle.  But  this  may  be 
the  sine  either  of  the  acute  angle  AB'C,  or  Q 

of  its  supplement,  the  obtuse  angle  ABC 
(Art.  27).    In  practice,  however,  there  will 

generally  be  some  circumstance  to  deter m-  

ine  whether  the  required  angle  is  acute  or  A.  B""*-  - — -  B' 
obtuse.  If  the  given  angle  is  obtuse,  there  can  be  no  ambi 
guity  in  the  solution,  for  then  the  remaining  angles  must  of 
course  be  acute. 

"Ex.  1.  In  a  triangle,  ABC,  there  are  given  AC,  458,  BO 
307,  and  the  angle  A,  28°  45',  to  find  the  other  parts. 

To  find  the  angle  B  : 

BC  :  sin.  A  :  :  AC  :  sin.  B. 

By  natural  numbers, 

307  :  .480989  :  :  458  :  .717566,  sin.  B,  the  arc  correspond 
ing  to  which  is  45°  51'  14",  or  134°  8'  46". 

This  proportion  is  most  easily  worked  by  logarithms,  thus 
BC,  307,  comp.,  7.512862 

:  sin.  A,  28°  45',  9.682135 

: :  AC,  458,  2.660865 

:  sin.  B,  45°  51'  14",  or  134°  8'  46",  9.8^5862. 
The  angle  ABC  is  134°  8'  46",  and  the  angie  AB'C,  45°  5 
14".     Hence  the  angle  ACB  is  17°  6'  14",  and  the  ang"  e  AOB 
105°  23'  46". 

To  find  the  side  AB  : 

sin.  A  :  CB  :  :  sin.  ACB  :  AB. 
By  logarithms, 

sin.  A,  28°  45V  comp.,  0.317865 

:  CB,  307,  2.487138 

: :  sin.  ACB,  17°  6'  14",        9.468502 
:  AB,  187.72,  2.273505. 

To  find  the  side  AB' : 

sin.  A  :  CB'  :  :  sin.  ACB  :  AB'. 
By  logarithms, 

sin.  A,  28°  45',  oomp.,  0.317865 

:  CB',  307,  2.487138 

: :  sin.  ACB',  105°  23  46"    9.984128 
:  AB',  615.36,  2.789131. 


V  L  A  N  E  .  T  R  .  G  0  N  0  M  E  T  R  V.  41 

Ex.  .$,  In  a  triangle,  ABC,  there  are  given  AB,  532,  BC; 
358,  and  the  angle  C,  107°  40','  to  find  the  other  parts. 

Ans.  A=39°52'52";  B=32°  27'.  8".  ;  AC-299,6. 

In  this  example  there  is  no  ambiguity,  because  the  givei. 
angle  is  obtuse. 

CASE  III. 

(55.)  Given  two  sides  and  the  included  angle,  to  find  tht 
third  side  and  the  remaining  angles. 

The  sum  of  the  required  angles  is  found  by  subtracting  the 
given  angle  from  180°.  The  difference  of  the  required  angles 
is  then  found  by  Theorem  II.  Half  the  difference  added  to 
half  the  sum  gives  the  greater  angle,  and,  subtracted,  gives 
the  less  angle.  The  third  side  is  then  found  by  Theorem  L 

Ex.  1.  In  the  triangle  ABC,  the  angle  A  is  given  53°  8'  , 
the  side  c,  420,  and  the  side  6,  535,  to  find  the  remaining  parts. 


The  sum  of  the  angles  B  +  C  =  lSO°-53°  8'  =  126°  52'. 
Half,  their  sum  is  63°  26'. 

Then,  by  Theorem  II., 

535+420  :  535-420  :  :  tang.  63C  26'  :  tang.  13°  32  25". 
which  is  half  the  difference  of  the  two  required  angles. 

Hence  the  angle  B  is  76°  58'  25",  and  the  angle  C,  49° 
53  35". 

To  find  the  side  a  : 

sin.  C  :  c  :  :  sin.  A  :  a=439.32. 

Ex.'  2.  Given  the  side  c,  176,  a,  133,  and  the  include';  &ngle 
B,  73°,  to  find  the  remaining  parts. 

Ans.,  6=187.022,  the  angle  C,  64°  9'  3",  and  A,  42°  W  57"', 


CASE  IY. 

(56.)  Given  the  three  sides,  to  find  the  angles. 

Let  fall  a  perpendicular  upon  the  longest  side  from  the  op- 
posite angle,  dividing  the  given  triangle  into  two  right-angled 
triangles.  The  two  segments  of  the  base  may  be  found  bv 
Theorem  III.  There  will  then  be  given  the  hypothenuso  and 
one  side  of  a  right-angled  triangle  to  find  the  anghs. 

Ex.  1.  In  the  triangle  ABC,  the  side  a  is  261,  the  sM*.  A« 
345,  and  c,  395.  What  are  the  angles  ? 

f^et  fall  the  perpendicular  CT)  upon  AB. 


42  TRIGONOMETRY 

Then,  ly  Theorem  III., 

AB  :  AC  +  CB  : :  AC-CB  :  AD~  DB; 
or  395  :  606  : :  84  :  128.87. 

Half  the  difference  of  the  segments  added  to  half  their  sun 
gives  the  greater  segment,  and  subtracted  gives  the  less  seg. 
ment. 

Therefore,   AD  is  261.935,   and  BD, 
133.065. 

Then,  in  each  of  the  right-angled  tri- 
angles, ACD,  BCD,  we  have  given  the  A  D  B 
hypothenuse  and  base,  to  find  the  angles  by  Case  II.  of  right- 
angled  triangles.  Hence 

AC : R  : : AD  :  cos.  A=40°  36'  13" ; 
BC  :  R  : :  BD  :  cos.  B=59°  20'  52' . 
erefore  the  angle  C=SO°  2'  55". 

Ex.  2.  If  the  three  sides  of  a  triangle  are  150,  140,  and  130, 
what  are  the  angles  ? 

Ans.,  67°  22'  48",  59°  29'  23",  and  53°  7'  49" 

Examples  for  Practice. 

1.  Given  two  sides  of  a  triangle,  478  and  567,  and  the  in 
eluded  angle,  47°  30',  to  find  the  remaining  parts. 

2.  Griven  the  angle  A,  56°  34',  the  opposite  side,  #,  735,  and 
the  side  #,  576,  to  find  the  remaining  parts. 

3.  G-iven  the  angle  A,  65°  40',  the  angle  B,  74°  20',  and  the 
side  a,  275,  to  find  the  remaining  parts. 

4.  G-iven  the  three  sides,  742,  657,  and  379,  to  find  the  an- 
gles. 

5.  Griven  the  angle  A,  116°  32',  the  opposite  side,  a,  492j 
and  the  side  c,  295,  to  find  the  remaining  parts. 

6.  Griven  the  angle  C,  56°  18',  the  opposite  side,  c,  184,  and 
the  side  &,  219,  to  find  the  remaining  parts 

This  problem  admits  of  two  answers. 

INSTRUMENTS  USED  IN  DRAWING. 

^57.)  The  fallowing  are  some  of  the  most  important  instru- 
ments used  in  drawing. 

I.  The  dividers  consist  of  two  legs,  revolving  upon  a  pivot 
at  oJie  extremity.  The  joints  should  be  composed  of  two  dif 


PLANE    T  R  i  G  o  N  o  M  K  i  R  Y. 


Icrent  metals,  of  unequal  hardness  r  one  part,  for  example,  ol 
steel,  and  the  other  of 
brass  or  silver,  in  order 
that  they  may  move  upon 
each  other  with  greater 
freedom.  The  points  should  he  of  tempered  steel,  and  when 
the  dividers  are  closed,  they  should  meet  with  great  exactness. 
The  dividers  are  often  furnished  with  various  appendages, 
which  are  exceedingly  convenient  in  drawing.  Sometimes  one 
of  the  legs  is  furnished  with  an  adjusting  screw,  by  which  a 
slow  motion  may  be  given  to  one  of  the  points,  in  which  case 
they  are  called  hair  compasses.  It  is  also  useful  to  have  a 
movable  leg,  which  may  be  removed  at  pleasure,  and  othe. 
parts  fitted  to  its  place ;  as,  for  example,  a  long  beam  for 
drawing  large  circles,  a  pencil  point  for  drawing  circles  with 
a  pencil,  an  ink  point  for  drawing  black  circles,  &c. 

(58.)  II.  The  parallel  rule  consists  of  two  flat  rules,  made 
of  wood  or  ivory,  and  connected  together  by  two  cross-bars  of 


equal  length,  and  parallel  to  each  other.  This  instrument  is 
useful  for  drawing  a  line  parallel  to  a  given  line,  through  a 
given  point.  For  this  purpose,  place  the  edge  of  one  of  the 
flat  rules  against  the  given  line,  and  move  the  other  rule  until 
:ts  edge  coincides  with  the  given  point.  A  line  drawn  along 
its  edge  will  be  parallel  to  the  given  line. 

(59.)  III.  The  protractor  is  used  to  lay  down  or  to  measure 
angles.  It  consists  of  a  sem- 
icircle, usually  of  brass,  and 
is  divided  into  degrees,  and 
sometimes  smaller  portions, 
the  center  of  the  circle  be- 
ing indicated  by  a  small 


notch. 


To  lay  down  an  angle  with  the  protractor,  draw  a  base  line, 
and  apply  to  it  the  edge  of  the  protractor,  so  that  its  center 
shall  fall  at  the  angular  point  Count  the  degrees  contained 


TRIGONOMETRY. 


in  the  proposed  angle  on  the  limb  of  the  circle,  and  mark  the 
extremity  of  the  arc  with  a  fine  dot.  Remove  the  instrument, 
and  through  the  dot  draw  a  line  from  the  angular  point ;  it 
will  give  the  angle  required.  In  a  similar  manner,  the  in- 
clination of  any  two  lines  may  be  measured  with  the  pro- 
tractor. 

(60.)  IV.  The  plane  scale  is  a  ruler,  frequently  two  feet  IP 
length,  containing  a  line  of  equal  parts,  chords,  sines,  tan 
gents,  &c.  For  a  ^cale  of  equal  parts,  a  line  is  divided  int; 
inches  and  tenths  of  an  inch,  or  half  inches  and  twentieths. 
When  smaller  fractions  are  required,  they  are  obtained  by 
means  of  the  diagonal  scale,  which  is  constructed  in  the  fol- 
g  planner.  Describe  a  square  inch,  ABC D,  and  divido 
4  -3  2  1  A.2.4.6.5  B 


DE 

jach  of  its  sides  into  ten  equal  parts.  Draw  diagonal  lines 
from  the  first  point  of  division  on  the  upper  line,  to  the  second 
on  the  lower ;  from  the  second  on  the  upper  line,  to  the  third 
on  the  lower,  and  so  on.  Draw,  also,  other  lines  parallel  t.3 
AB,  through  the  points  of  division  of  BC.  Then,  in  the  trian- 
gle ADE,  the  base,  DE,  is  one  tenth  of  an  inch;  and,  since 
the  line  AD  is  divided  into  ten  equal  parts,  and  through  the 
points  of  division  lines  are  drawn  parallel  to  the  base,  forming 
nine  smaller  triangles,. the  base  of  the  least  is  one  tenth  of  DE, 
that  is,  .01  of  an  inch ;  the  base  of  the  second  is  .02  of  an  inch ; 
the  third,  .03,  and  so  on.  Thus  the  diagonal  scale  furnishes 
us  hundredths  of  an  inch.  To  take  off  from  the  scale  a  line 
of  given  length,  as,  for  example,  4.45  inches,  place  one  foot  of 
the  dividers  at  F,  on  the  sixth  horizontal  line,  and  extend  the 
other  foot  to  Gr,  the  fifth  diagonal  line. 

A  half  inch  or  less  is  frequently  subdivided  in  the  samo 
manner. 

(61.)  A  line  of  chords,  commonly  marked  CHO.,  is  found  011 
most  plane  scales,  and  is  useful  in  setting  off  angles.  To  form 
this  line,  describe  a  circle  with  any  convenient  radius,  and  di- 
the  circumference  into  degrees.  Let  the  length  of  the 


PLANE    TRIGONOMETRY. 


chords  foi  every  degree  of  the  quadrant  be  determined  and  laid 
i>ff  on  a  scale  :  this  is  called  a  line  of  chords. 

Since  the  chord  of  60°  is  equal  to  radius,  in  order  to  lay 


down  an  angle,  we  take  from  the  scale  the  chord  of  60°,  and 
with  this  radius  describe  an  arc  of  a  circle.  Then  take  from 
the  scale  the  chord  of  the  given  angle,  and  set  it  off  upon  the 
former  arc.  Through  these  two  points  of  division  draw  lines 
to  the  center  of  the  circle,  and  they  will  contain  the  required 
angle. 

The  line  of  sines,  commonly  marked  SIN.,  exhibits  the  lengths 
of  the  sines  to  every  degree  of  the  quadrant,  to  the  same  ra- 
dius as  the  line  of  chords.  The  line  of  tangents  and  the  line 
of  secants  are  constructed  in  the  same  manner.  .  Since  the  sine 
of  90°  is  equal  to  radius,  and  the  secant  of  0°  is  the  same,  the 
graduation  on  the  line  of  secants  begins  where  the  line  of  sine* 
ends. 

On  the  back  side  of  the  plane  scale  are  often  found  lines  rep 
icsenting  the  logarithms  of  numbers,  sines,  tangents,  &c.  Thin 
is  called  Grunter's  Scale. 

(62.)  Y.  The  Sector  is  a  very  convenient  instrument  in 
drawing.  It  consists  of 
two  equal  armc,  mova- 
ble about  a  pivot  as  a 
center,  having  several 
scales  drawn  on  the 
faces,  some  single,  oth- 
ers double.  The  single  scales  are  like  those  upon  a  common 
Gunter's  scale.  The  double  scales  are  those  which  proceed 
from  the  center,  each  being  laid  twice  on  the  same  face  of  the 
instrument,  viz.,  once  on  each  leg.  The  double  scales  are  a 
scale  of  lines,  marked  Lin.  or  L. ;  the  scale  of  chords,  sines, 
&c.  On  each  arm  of  the  sector  there  is  a  diagonal  line,  which 
diverges  from  the  central  point  like  the  radius  of  a  circle,  and 
these  diagonal  lines  are  divided  into  equal  parts. 

The  advantage  of  the  sector  is  to  enable  us  to  di?k\v'a'lin« 


46  TfllGONOMETRv 

upon  paper  to  any  scale ;  as,  for  example,  a  scale  of  6  feel  to 
the  inch.  For  this  purpose,  take  an  inch  with  the  dividers 
from  the  scale  of  inches  ;  then,  placing  one  foot  of  the  dividers 
at  6  on  one  arm  of  the  sector,  open  the  sector  until  the  othe? 
foot  reaches  to  the  same  number  on  the  other  arm.  Now,  re- 
garding the  lines  on  the 
sector  as  the  sides  of  a 
triangle,  of  which  the 
lino  measured  from  6  on 
one  arm  to  6  on  the  oth- 
er arm  is  the  base,  it  is 
plain  that  if  any  other  line  be  measured  across  the  angle  of 
the  sector,  the  bases  of  the  triangles  thus  formed  will  be  pro- 
portional to  their  sides.  Therefore,  a  line  of  7  feet  will  be  rep- 
resented by  the  distance  from  7  to  7,  and  so  on  for  other  lines. 

The  sector  also  contains  a  line  of  chords,  arranged  like  the 
line  of  equal  parts  already  mentioned.  Two  lines  of  chords 
are  drawn,  one  on  each  arm  of  the  sector,  diverging  from  the 
central  point.  This  double  line  of  chords  is  more  convenient 
than  the  single  ono  upon  the  plane  scale,  because  it  furnishes 
chords  to  any  radius.  If  it  be  required  to  lay  down  any  angle, 
as,  for  example,  an  angle  of  25°,  describe  a  circle  with  any 
convenient  radius.  Open  the  sector  uo  that  the  distance  from 
CO  to  60,  on  the  line  of  chords,  shall  be  equal  to  this  radius. 
Then,  preserving  the  same  opening  of  the  sector,  place  one  foot 
of  the  dividers  upon  the  division  25  on  one  scale,  and  extend 
the  other  foot  to  the  same  number  upon  the  other  scale :  this 
distance  will  be  the  chord  of  25  degrees,  which  must  be  set  off 
upon  the  circle  first  described. 

The  lines  of  sines,  tangents,  &c.,  are  arranged  in  the  same 
manner. 

(63.)  By  means  of  the  instruments  now  enumerated,  all  the 
cases  in  Plane  Trigonometry  may  be  solved  mechanically. 
The  sides  and  angles  which  are  given  are  laid  down  accord- 
ing to  the  preceding  directions,  and  the  required  parts  are  then 
measured  from  the  same  scale.  The  student  will  do  well  to 
exercise  himself  upon  the  following  pr  ?blems  : 

I.  Given  the  angles  and  one  side  oj  a  triangle,  to  find,  by 
construction,  the  other  two  sides 


PLANE    TRIGONOMETRY.  47 

Draw  an  indefinite  straight  line,  and  from  thu  scale  ui  equa. 
parts  lay  off  a  portion,  AB,  equal  to  the  given  side.  From 
each  extremity  lay  off  an  angle  equal  to  one  of  the  adjacent  an- 
gles, by  means  of  a  protractor  or  a  scale  of  chords.  Extend 
the  two  lines  till  they  intersect,  and  measure  their  lengths  upon 
the  same  scale  of  equal  parts  which  was 
used  in  laying  off  the  base. 

Ex.  1.  Given  the  angle  A,  45°  30',  the 
angle  B,  35°  20',  and  the  side  AB,  43° 
rods,  to  construct  the  triangle,  and  find 
the  lengths  of  the  sides  AC  and  BC. 

The  triangle  ABC  may  be  constructed  of  any  dimensions 
whatever ;  all  which  is  essential  is  that  its  angles  be  made 
equal  to  the  given  angles.  We  may  construct  the  triangle 
upon  a  scale  of  100  rods  to  an  inch,  in  which  case  the  side  AB 
will  be  represented  by  4.32  inches ;  or  we  may  construct  it 
upon  a  scale  of  200  rods  to  an  inch ;  that  is,  100  rods  to  a  half 
inch,  which  is  very  conveniently  done  from  a  scale  on  which 
a  half  inch  is  divided  like  that  described  in  Art.  60 ;  or  we 
may  use  any  other  scale  at  pleasure.  It  should,  however,  be 
remembered,  that  the  required  sides  must  be  measured  upcn 
the  same  scale  as  the  given  sides. 

Ex.  2.  Given  the  angle  A,  48°,  the  angle  C,  113°,  and  the 
side  AC,  795,  to  construct  the  triangle. 

II.  Given  twi>  sides  and  an  angle  opposite  one  of  them,  tn 
find  the  other  parts. 

Draw  the  side  which  is  adjacent  to  the  given  angle.  From 
one  end  of  it  lay  off  the  given  angle,  and  extend  a  line  indefin- 
itely for  the  required  side.  From  the  other  extremity  of  the 
first  side,  with  the  remaining  given  side  for  radius,  describe 
an  arc  cutting  the  indefinite  line.  The  point  of  intersection 
will  determine  the  third  angle  of  the  triangle. 

Ex.  1.  Given  the  angle  A,  74°  45',  the  side  AC,  432,  and 
the  side  BC,  475,  to  construct  the  triangle,  and  find  the  other 
parts. 

Ex.  2.  Given  the  angle  A,  105°,  the  side  BC,  498,  and  the 
side  AC,  375,  to  construct  the  triangle. 

III.  Given  two  sides  and  the  included  angle,  to  find  the 
itlwr  parts. 


*8  TRIG  GNOME  TR\ 

Draw  one  of  the  given  sides.  From  one  end  of  it  lay  off  the 
given  angle,  and  draw  the  other  given  side,  making  the  re- 
quiied  angle  with  the  first  side.  Then  connect  the  extremities 
of  the  two  sides,  and  there  will  be  formed  the  triangle  required 

Ex.  1.  G-iven  the  angle  A,  37°  25',  the  side  AC,  675,  and 
the  side  AB,  417,  to  construct  the  triangle,  and  find  the  othci 
parts. 

Ex.  2.  Given  the  angle  A,  75°,  the  side  AC,  543,  and  the 
side  AB,  721,  to  construct  the  triangle. 

IV.  Given  the  three  sides,  to  find  the  angles. 

Draw  one  of  the  sides  as  a  "base ;  and  from  one  extremity 
of  the  base,  with  a  radius  equal  to  the  second  side,  describe 
an  arc  of  a  circle.  From  the  other  end  of  the  base,  with  a 
radius  equal  to  the  third  side,  describe  a  second  arc  intersect- 
ing the  former ;  the  point  of  intei  section  will  be  the  third  an- 
gle of  the  triangle. 

Ex.  1.  Given  AB,  678,  AC,  598,  and  BC,  435,  to  find  tho 
angles. 

Ex.  2.  Given  the  three  sides  476,  287,  and  354,  to  find  tho 
angles. 

Values  of  the  Sines,  Cosines,  Sfc.,  of  certain  Angles 
(64.)  We  propose  now  to  examine  the  changes  which  ths 
tines,  cosines,  &c.,  undergo  in  the  dif- 
ferent quadrants  of  a  circle.  Draw 
two  diameters,  AB,  DE,  perpendicu- 
lar to  each  other,  and  suppose  one  of 
them  to  occupy  a  horizontal  position, 
the  other  a  vertical.  The  angle  ACT) 
is  called  the  first  quadrant,  the  angle 
DCB  the  second  quadrant,  the  angle 
BCE  the  third  quadrant,  and  the  an- 
gle EGA  the  fourth  quadrant;  that  is,  the  first  quadrant  is 
above  the  horizontal  diameter,  and  on  the  right  of  the  vertical 
diameter  ;  the  second  quadrant  is  above  the  horizontal  diame- 
2cr,  and  on  the  left  of  the  vertical,  and  so  on. 

Suppose  one  extremity  of  the  arc  remains  fixed  in  A,  while 
the  other  extremity,  marked  F, runs  round  the  entire  circum- 
foronce  in  the  direction  ADBE. 


PLANE    T  R  i  G  o  N  o  M  E  T  R  Y.  49 

Whm  the  point  F  is  at  A,  or  when  the  arc  AF  is  zero,  the 
tine  is  zero.  As  the  point  F  advances  toward  D,  the  sine  in- 
creases ;  and  when  the  arc  AF  becomes  45°,  the  triangle  CFQ 
being  isosceles,  we  have  FGr  :  CF  :  :  1  :  y/2  (Geom.,  Prop.  11, 
Cor.  3,  B.  IV.)  ;  or  sin.  45°  :  R  :  :  1  :  ^2. 

[fence,  sin.  45D=  —  -=£Rv/2. 

\/Z 

The  sine  of  30°  is  equal  to  half  radius  {Art.  22).  Also,  since 
sin.  A=  VR2  —  cos.  2A,  the  sine  of  60°,  which  is  equal  to  the  co 


sine  of  30°,  =  Vll'-±R*=  VfRJR  V3. 

The  arc  AF  continuing  to  increase,  the  sine  also  increase 
till  F  arrives  at  D,  at  which  point  the  sine  is  equal  to  the  ra 
dius  ;  that  is,  the  sine  of  90°  =  R. 

As  the  point  F  advances  from  D  toward  B,  the  sines  dimin- 
ish, and  become  zero  at  B  ;  that  is,  the  sine  of  180°  =  0. 

In  the  third  quadrant,  the  sine  increases  again,  becomes 
iqual  to  radius  at  E,  and  is  reduced  to  zero  at  A. 

(65.)  "When  the  point  F  is  at  A,  the  cosine  is  equal  to  ra- 
dius. As  the  point  F  advances  toward  D,  the  cosine  decreases, 
and  the  cosine  of  45°  =  sine  45°  —  JR  v/2.  The  arc  continuing 
to  increase,  the  cosine  diminishes  till  F  arrives  at  D,  at  which 
point  the  cosine  becomes  equal  to  zero.  The  cosine  in  the  sec- 
ond quadrant  increases,  and  becomes  equal  to  radius  at  B  ;  in 
the  third  quadrant  it  decreases,  and  becomes  zero  at  E  ;  in  the 
fourth  quadrant  it  increases  again,  and  becomes  equal  to  ra- 
dius at  A. 

(66.)  The  tangent  begins  with  zero  at  A,  increases  with  the 
arc,  and  at  45°  becomes  equal  to  radius.  As  the  point  F  ap- 
proaches D,  the  tangent  increases  very  rapidly  ;  and  when  the 
difference  between  the  arc  and  90°  is  less  than  any  assignable 
quantity,  the  tangent  is  greater  than  any  assignable  quantity 
Hence  the  tangent  of  90°  is  said  to  be  infinite. 

In  the  secoud  quadrant  the  tangent  is  at  first  infinitely  great, 
and  rapidly  diminishes  till  at  B  it  is  reduced  to  zero.  In  the 
third  quadrant  it  increases  again,  becomes  infinite  at  E,  and  is 
reduced  to  zero  at  A. 

The  cotangent  is  equal  to  zero  at  D  and  E,  and  is  infinite  at 
A  and  B. 

/f>7.)  The  secant  begins  with  radius  at  A,  increases  through 


50  T  R  I  G  0  N  0  M  E  T  R  "i . 

the  first  quadrant,  and  becomes  infinite  at  D  :  diminishes  in 

the  second  quadrant,  till  at  B  it  is 

equal  to  the  radius ;  increases  again 

in  the  third  quadrant,  and  becomes 

infinite  at  E  ;  decreases  in  the  fourth 

quadrant,  and  becomes  equal  to  the   B 

radius  at  A. 

The  cosecant  is  equal  to  radius  at 
D  and  E,  and  is  infinite  at  A  and  B. 

(68.)  Let  us  now  consider  the  al- 
gebraic  signs  by  which  these  lines  are  to  be  distinguished.  Jn 
the  first  and  second  quadrants,  the  sines  fall  above  the  diame- 
ter AB,  while  in  the  third  and  fourth  quadrants  they  fall  be- 
low. This  opposition  of  directions  ought  to  be  distinguished 
by  the  algebraic  signs  ;  and  if  one  of  these  directions  is  re- 
garded as  positive,  the  other  ought  to  be  considered  as  nega- 
tive. It  is  generally  agreed  to  consider  those  sines  which  fall 
above  the  horizontal  diameter  as  positive  ;  consequently,  those 
which  fall  below  must  be  regarded  as  negative.  That  is,  the 
sines  are  positive  in  the  first  and  second  quadrants,  and  nega- 
tive in  the  third  and  fourth. 

In  the  first  quadrant  the  cosine  falls  on  the  right  of  DE, 
but  in  the  second  quadrant  it  falls  on  the  left.  These  two  linos 
should  obviously  have  opposite  signs,  and  it  is  generally  agreed 
to  consider  those  which  fall  to  the  right  of  the  vertical  diam- 
eter as  positive  ;  consequently,  those  which  fall  to  the  left  must 
be  considered  negative.  That  is,  the  cosines  are  positive  in 
the  first  and  fourth  quadrants,  and  negative  in  the  second  and 
third. 

(69.)   The  signs  of  the  tangents  are  derived  from  those  of 

the  sines  and  cosines.     For  tang.  = '  (Art.  28).     Henco 

when  the  sine  and  cosine  have  like  algebraic  signs,  the  tan- 
gent will  be  positive ;  when  unlike,  negative.  That  is,  the  tan 
gent  is  positive  in  the  first  and  third  quadrants,  and  negative 
in  the  second  and  fourth. 

T>2 

Also,  cotangent  =- (Art.  28) ;  hence  the  tangent  and 

cotangent  have  always  the  same  sign 


ELANE   TRIGONOMETRY.  51 

T>2 

We  have  seen  that  sec,  = — - ;  hence  the  secant  must  hav« 

cos. 

the  same  sign  as  the  cosine. 

TJ8 

A] so,  cosec.  =— — :  hence  the  cosecant  must  have  the  same 
sin. ' 

sign  as  the  sine. 

(70.)  The  preceding  results  are  exhibited  in  the  following 
Uoles,  which  should  be  made  perfectly  familiar : 

First  quad.     Second  quad.     Third  quad.    Fourth  quad 

fiine  and  cosecant,  -t-  + 

Cosine  and  secant,  -f-  + 

Tangent  and  cotangent,     -f  -f 


Sine, 

Cosine, 

Tangent, 

Cotangent, 

Secant, 

Cosecant, 

(71.)  In  Astronomy  we  frequently  have  occasion  to  considei 
greater  than  360°.  But  if  an  entire  circumference,  or  any 
number  of  circumferences,  be  added  to  any  arc,  it  will  termin- 
ate in  the  same  point  as  before.  Hence,  if  C  represent  an  en- 
tiro  circumference,  or  360°,  and  A  any  arc  whatever,  we  shall 
have 

sin,  A=sin.  (C-fA)=sin.  (2C+A)=sin.  (3C+A)  =  ,  &c. 

The  same  is  true  of  the  cosine,  tangent,  &c. 

We  generally  consider  those  arcs  as  positive  which  are  esti- 
mated from  A  in  the  direction  ADBE.  If,  then,  an  arc  were 
estimated  in  the  direction  AEBD,  it  should  be  considered  as 
negative ;  that  is,  if  the  arc  AF  be  considered  positive,  AH 
must  be  considered  negative.  But  the  latter  belongs  to  the 
fourth  quadrant;  hence  its  sine  is  negative.  Therefore,  sin 
(— A)  =  —  sin.  A. 

The  cosine  CG  i.  the  same  for  both  the  arcs  AF  and  AH- 
Hence,  cos.  (— A)=cos.  A 

AJso,  tang.  (— A)  =  — tang.  A. 

A  nd  cot,  ( — A)  =  —  cot.  A 


0° 

90° 

180° 

270° 

360- 

0 

•f'B 

0 

-R 

0 

-fR 

0 

•-R 

0 

+R 

0 

GO 

0 

GO 

0 

it,               00 

0 

00 

0 

00 

+R 

GO 

-R 

GO 

+R 

00 

+R 

GO 

-R 

GO 

TRIGONOMETRY 


TRIGONOMETRICAL  FORMULAE. 

(72.)  Expressions  for  the  sine  and  cosine  of  the  sum  ana 
difference  of  two  arcs. 

Let  AB  and  BD  represent  any  two  given  arcs ;  take  BE 
equal  to  BD  :  it  is  required  to  find  an         D 
expression  for  the  sine  of  AD,  the  sum, 
and  of  AE,  the  difference  of  these  arcs. 

Put  AB=a,  and  BD=6;  then  AD= 
a -f b,  and  AE  =  a—b.  Draw  the  chord 
DE,  and  the  radius  CB,  which  may  be 
represented  by  R.  Since  DB  is  by 
construction  equal  to  BE,  DF  is  equal 
to  FE,  and  therefore  DE  is  perpendic- 
ular to  CB.  Let  fall  the  perpendicular  EG,  BH,  FI,  and  DK 
upon  AC,  and  draw  EL,  FM  parallel  to  AC. 

Because  the  triangles  BCH,  I1  CI  are  dnular,  we  have 
CB  :  CF  •  :  BH  :  FI ;  or  R  :  cos.  b  :  :  sin.  a  :  FI. 


Therefore, 


sin.  a  cos.  b 

~~~- 


Also,  CB  :  CF  :  :  CH  :  CI ;  or  R  :  cos.  b  :  :  cos.  a  :  CI. 


Therefore, 


CI=- 


cos.  a  cos,  b 


The  triangles  DFM,  CBH,  having  their  sides  perpendicula! 
each  to  each,  are  similar,  and  give  the  proportions 

CB  :  DF  :  :  CH  :  DM  ;  or  R  :  sin.  b  :  :  cos.  a  :  DM. 


Hence 


.____     cos.  a  sin.  b 
DM=  ---  „  --  . 


Also,  CB  :  DF  :  :  BH  :  FM  ;  or  R  :  sin.  b  :  :  sin.  a  :  FM 

sin.  a  sin.  b 
[fence 


But 

and 

Also, 

and 


Hence, 


FI+DM=DK  =sin.  (a+bj  ; 
CI-FM=CK  =cos.  (a+b). 
FI-FL  =Ea=sin.  (a-b)  ; 
CI+EL  =  Ca=cos.  (a-b). 

T.     sin.  a  cos.  b+cos.  a  sin.  b 
sm.  (a+  b)=—  —  ^  —  — 

cos.  a  cos  b—  sin.  a  sin.  b 
cos  (a+b)=  R-  - 


... 
(1) 


PLANE    TRIGONOMETRY.  53 

,  .     sin.  a  cos.  b—  cos.  a  sin.  b 
sm.  (a-b}  =  -  -g—  -   (3) 

cos.  a  cos.  6+sin.  a  sin.  & 
«*.(*-*)=  -^—  -  (4) 

(70.)  Expressions  for  the  sine  and  cosine  of  a  double  are. 
If,  in  the  formulas  of  the  preceding  article,  we  make  b=a 
fco  first  and  second  will  become 

2  sin.  a  cos.  a 


sm.    a= 


cos.  2a= 


=r 
it 

cos.  #  —  sn. 


Making  radius  equal  to  unity,  and  substituting  the  values  oi 
sin.  #,  cos.  a.  &c.,  from  Art.  28,  we  obtain 

2  tans,  a 


sn.    a— 


rt 
cos.  2^= 


1 4- tang.  V 
1— tanff.  2# 


^  . 

1+tang.  a 

(74.)  Expressions  for  the  sine  and  cosine  of  half  a  give* 
arc. 

Tf  we  put  \a  for  a  in  the  preceding  equations,  we  obtain 

2  sin.  \a  cos.  A# 
sm.  a=          —  g—      —  , 

cos.  ^a~  ?:n.  2Aa 


cos.  a=  -  -  —  ^  —       —. 
K 

We  may  also  find  the  sine  and  cosine  of  ^  in  terms  of  a. 
Since  the  sum  of  the  squares  of  the  sine  and  cosine  is  equal 
;  the  square  of  radius,  we  have 

cos.  2^#+sin.  2J#  =  R2. 
And,  from  the  preceding  equation, 

cos.  *i«—  sin.  S^#=R  cos.  a. 
If  we  subtract  one  of  these  from  the  other,  we  have 

2  sin.  9J#=R2—  R  cos.  &. 
And,  adding  the  same  equations, 

2  cos.  2^<z=R2+R  cos.  a. 
Hence,  sin.  |a=  v^R2—  ^R  cos.  a; 


cos.      =         ,+,  cos.  a. 

(75.)  Expressions  for  the  products  of  sines  and  cosines 
By  adding  and  subtracting  the  formulas  of  Art.  72,  we  obtain 


M  TRIGONOMETRY. 

sin.  (a+b)+&in.  («—  M=-    sin.  «  cos.  b. 

JK 
o 

sin.  (a+&)—  sin.  (#—&)=—  cos.  a  sin.  ft; 

o 
cos.  (a-\-b)+cos.  (a—b)=^-  cos.  a  cos.  b  ; 

2 
cos.  (a—  b)—  cos.  (a+^)=^-  sin.  a  sin.  b. 

If,  in  these  formulas,  we  make  a+b=  A,  and  a—  6—H  ;  that 
N,  a=J(A+  B),  and  &=J(A-B),  we  shall  have 

2 
sin.  A+sin.  B=^-  sin.  J(A+  B)  cos.  J(A-B)     (1) 

2 
sin.  A-sin.  B=j£  sin.  £(A-B)  cos.  i(A-f  B)     (2) 

2 
cos.  A+cos.  B=^-  cos.  J(A-f-B)  cos.  |(A-B)     (3) 

2 
cos.  B-eos.  A=p  sin.  |(A-fB)  sin.  J{A-B)     (4) 

(76.)   Dividing  formula  (1)  by  (2),  and  considering  that 

sin.  a    tang,  a 

-  =  —  ^  —  (Art-  28),  we  have 

cos.  a        R       v 


sin.  A+sin.  B_sin.  ^(A+B)  cos.  ^(A-B)_tang.  J(A-f  B)^ 
sin.  A-sin.  B~sin.  i(A-B)  cos.  i(A4-B)""tang.  i(A-B); 

that  is, 

The  sum  of  the  sines  of  two  arcs  is  to  their  difference,  as 

the  tangent  of  half  the  sum  of  those  arcs  is  to  the  tangent  of 

naif  their  difference. 

.Dividing  formula  (3)  by  (4),  and  considering  that  ~r~==~:s~ 

p 
=*--  —  (Art.  28),  we  have 

**   5* 

cos.  A+cos.  B_cos.  ^(A+B)  cos.  ^(A-B)_  cot. 


cos.  B-cos.  A~~sin.  i(A-f  B)  sin.    (A~B~tang4(A-B) 
that  is, 

The  sum  of  the  cosines  of  two  arcs  is  to  their  difference,  as 
the  cotangent  of  half  the  sum  of  those  arcs  is  to  the  tang*  ft 
of  half  their  difference. 


PLANE    TRIGONOMETRY.  55 

From  the  first  formula  of  Art.  74,  by  substituting  A-f  B  foi 
A.  we  have 


Dividing  formula  (1),  Art.  75,  by  this,  we  obtain 
sin.  A+sin.  B_sin.  -j(A  +  B)  cos.  ^(A-B)_cos.  j(A-K)  t 
sin.  (A-f  B)  "sin.  |(A+B)  cos.  i(A+B)""oos"."J(A+B)  : 
I  hat  is, 

The  sum  of  the  sines  of  two  arcs  is  to  the  sine  of  their 
sum,  as  the  cosine  of  half  the  difference  of  those  arcs  is  to 
the  cosine  of  half  their  sum. 

If  we  divide  equation  (1),  Art.  72,  by  equation  (3),  we  sha  1 
have 

sin.  (a-\-b)     sin.  a  cos.  b+cos.  a  sin.  b 
sin.  (a  —  b)     sin.  a  cos.  b—  cos.  a  sin.  b' 
By  dividing  both  numerator  and  denominator  of  the  second 

tang.         sin. 

member  by  cos.  a  cos.  b,  and  substituting  —  =:  —  for  -  ,  we  ob- 

R  cos. 

sin.  (a-\-b)     tang,  0  +  tans'.  b 
tain  -.  —  7  -  rr=r-  —  —  /  /  that  is, 

sin.  (a  —  b)     tang,  a  —  tang,  b 

The  sine  of  the  sum  of  two  arcs  is  to  the  sine  of  their  dif- 
ference^ as  the  sum  of  the  tangents  of  those  arcs  is  to  the 
difference  of  the  tangents. 

From  equation  (3),  Art.  72,  by  dividing  each  member  by  co<* 
a  cos.  b,  we  obtain 

sin.  (a—b)  _sin.  a  cos.  b—  cos.  a  sin.  &__tang.  a—  tang,  b 

cos.  a  cos.  b  R  cos.  a  cos.  b  R2 

that  is, 

The.  sine  of  the'  difference  of  two  arcs  is  to  the  product  of 
their  cosines,  as  the  difference  of  their  tangents  is  to  the 
square  of  radius. 

(77.)  Expressions  for  the  tangents  of  arcs. 

If  we  take  the  expression  tang.  (a+b)=  -  ~  —  rr~-  (Art. 

cos.  (a+b)     v 

28),  and  substitute  for  sin.  (a-\-b)  and  cos.  (a+b)  their  values 
given  in  Art.  72,  we  shall  find 

..     R  (sin.  a  cos.  &-fcos.  a  sin.  b) 
tang.  (a+b)~-  —L  -  :  ---  :  —  7-/. 

cos.  a  cos.  b—  sin  a  sm.  b 


TRIGONOMETRY 


.  oos.  a  tang,  a  .  cos.  &  tang,  b 

Britain  a~  -  =--   —  ,  andsm.  &  =  —    -^-     —  (Art,28j 

If  we  substitute  these  values  in  the  preceding  equation,  and 
divide  all  the  terms  by  cos.  a  cos.  b,  we  shall  have 

7.     R2  (tang,  a  +  tang,  b) 
tang,  (a  f£)=_  ' 


R  —  tang,  a  tang,  b 
In  like  manner  \ve  shall  find 

Ra  (tang,  a  -tang.  6) 


Suppose  6=  a,  then 

2R3  tang.  a 


tang.  20  =  ^ 
' 


Suppose  b=2a,  then 


. 

R'-tan/a 


R2  (tans?,  a  +  tan<?.  2a) 
; 


tang.  3a— 

"—  tang,  a  tang.  2a 

In  the  same  manner  we  find 

cot.  a  cot.  b  —  Ra 

cot.  (a  +  b)  -•—    --f—.  —       —  , 

cot.  &+cot.  a   ' 

cot.  a  cot.  &+Ra 
cot.  (a  —  b)=  --  —-J  —        —  . 
cot.  b—  cot.  a 

(78.)  When  the  three  sides  of  a  triangle  are  givenv  the  an- 
gles may  be  found  by  the  formula 


where  S  represents  half  the  sum  of  the  sides  a,  6,  and  c. 

Demonstration. 

Let  ABC  be  any  triangle  ;  then  (Geom.,  °rop.  12,  B.  IV.), 
BCa=AB'-fAC9-2ABxAD. 

'     AB'-fAC'-BCr; 
Hence,       AD=       ~2AB~   ~~' 

But  in  the  right-angled  triangle  ACD, 
we  have 

R  :  AC  :  :  cos.  A  :  AP 

ir  A     RxAP 

Hence,  cos.  A=  —  7-7=  —  ; 

AL/ 

ar.  by  substituting  the  value  of  AD, 

ABa+ACa-BOa 


cos.     ^ 


SABxAC 


Jt*  L  A  K  E     T  R  I  G  0  N  O  M  K  T  R  \r.  57 

Let  a,  b,  .:  represent  the  sides  opposite  the  angles  A,  B,  C  , 

p     V    S-a? 
f.n  cos.  A=KX 


By  Art.  74,  we  have  2  sin.  2JA=IT-R  cos.  A. 
Substituting  for  cos.  A  its  value  given  above,  we  obi*  '.n 


2bc 
Put  S  =  J(a-f-#  +  c),  and  we  obtain,  after  reduction, 


Fn  the  same  manner  we  find 


3— a)  (S  —  f) 
ac 


Kx.  1.  What  are  the  angles  of  a  plane  triangle  whose  sidea 
are  432,  543,  and  654  ? 

Here  8=814.5;  S-£  =  382.5  ;  S-c=271.5. 

log.  382.5  2.582631 

log.  271.5  2.433770 

log.  b,  432  comp.    7.364516 

log.  r,  543  comp.    7.2G5200 

2)  19.646117 

sin.  -|A,  41°  42'  36J".  9.823058. 

Angle  A=83°  25'  13". 

In  a  similar  manner  we  find  the  angle  B=41°  0'  39  ',  and 
the  angle  0  =  55°  34'  8". 

Ex.  2.  What  are  the  angles  of  a  plane  triangle  whose  side* 
are  245,  219,  and  91  ? 

(79.)  On  the  computation  of  a  table  of  sines,  cosines,  &*c. 
In  computing  a  table  of  sines  and  cosines,  we  begin  with 
finding  the  sine  and  cosine  of  one  minute,  and  thence  deduce 
the  sines  and  cosines  of  larger  arcs.  The  sine  of  so  small  an 
angle  as  one  minuto  is  nearly  equal  to  the  corresponding  aio 
The  radius  being  taken  as  unity,  the  semicircumferenco  i> 


5b  TRIGONOMETRY. 

known  to  be  3.14159.  This  being  divided  successively  by  ISC 
and  60,  gives  .0002908882  for  the  arc  of  one  minute,  which 
may  be  regarded  as  the  sine  of  one  minute. 

The  cosine  of  1'=  /I^sin?= 0.9999999577. 
The  sines  of  very  small  angles  are  nearly  proportional  to  the 
angles  themselves.  We  might  then  obtain  several  other  sines 
by  direct  proportion.  This  method  will  give  the  sines  correct 
to  five  decimal  places,  as  far  as  two  degrees.  By  the  follow- 
ing method  they  may  be  obtained  with  greater  accuracy  for 
the  entire  quadrant. 

By  Art.  75,  we  have,  by  transposition, 

sin.  (a+b)=2  sin.  a  cos.  6— sin.  (&— 6), 
cos.  (a+b)=2  cos.  a  cos.  b— cos.  (a— b). 
If  we  make  a=b,  26,  36,  &c.,  successively,  we  shall  hav«? 
sin.  26=2  sin.  b  cos.  b  ; 
sin.  36=2  sin.  2b  cos.  b— sin.  b 
sin.  46=2  sin.  36  cos.  b— sin.  26, 

&c.,  &c. 

cos.  26=2  cos.  6  cos.  6—1 ; 
cos.  36=2  cos.  26  cos.  6— cos.  6  : 
cos.  46=2  cos.  36  cos.  6— cos.  26, 

&c.,  &c, 

Wl  jnce,  making  6=1',  we  have 

sin.  2' =2  sin.  V  cos.  1'  =.000582 

sin.  3'=2  sin.  2'  cos.  I'-sin.  1'=.000873; 
sin.  4'=2  sin.  3'  cos.  I'-sin.  2'=.001164, 

&c.,  &c. 

cos.  2'=2  cos.  V  cos.  1'-  1  =0.999999 ; 
cos.  3'  =  2  cos.  2'  cos.  I'-cos.  l'=0.999999  ; 
cos.  4' =2  cos.  3'  cos.  I'-cos.  2' =0.999999, 

&c.,  &c. 

The  tangents,  cotangents,  secants,  and  cosecants  aio  ca^i!]? 
iari"«wJ  from  the  sines  and  Dosines.     Thus, 

sin.  V  cos.  V 

tang.  r= T-  ;  cot.  l'=- — r  ; 

cos.  1'  sin.  1  ' 

1  1 

sec.  1'— -r-  j          cosec.  1  =- — ~ ; 

cos.  1  '  am    1' ' 


)U  OK  III. 

MENSURATION  OF  SURFACES. 

(80.)  THE  arsa  of  a  figure  is  the  space  contained  within  thf 
lino  or  lines  by  which  it  is  bounded.  This  area  is  determined 
by  finding  how  many  times  the  figure  contains  some  other  sur- 
face, which  is  assumed  as  the  unit  of  measure.  This  unit  is 
commonly  a  square ;  such  as  a  square  inch,  a  square  foot,  a 
square  rod,  &c. 

The  superficial  unit  has  generally  the  same  name  as  the 
linear  unit,  which  forms  the  side  of  the  square.     Thus, 
the  side  of  a  square  inch  is  a  linear  inch  ; 
"      "    of  a  square  foot  is  a  linear  foot ; 
u      "     of  a  square  yard  is  a  linear  yard,  &o. 

There  are  some  superficial  units  which  have  no  correspond- 
ing linear  units  of  the  same  name,  as,  for  example,  an  acre. 

The  following  table  contains  the  square  measures  in  com- 
mon use : 

Table  of  Square  Measures. 

Sq.  Inches.  Sq.  Feel. 

144  =  1  5^.  yards 

1296=  9  -  1       Sq.Rods. 

39204-          272|=          30£=          1    5.c;,, 
627264=         4356  =         484  =         16=       1     AM 
6272640=       43560  =       4840  =       160=     10=     1   M 
4014489600=27878400  =3097600  =102400=6400=640=1 

PROBLEM  I. 
(81.)  To  find  the  area  of  a  parallelogram. 

RULE  I. 

Multiply  the  base  by  the  altitude. 

For  Ihe  demonstration  of  this  rule,  see  Geometry,  Prop  5 
B.  IV 


60  TRIGONOMETRY. 

Ex.  1.  What  is  the  area  of  a  parallelogram  whoso  base  ia 
17.5  rods,  and  the  altitude  13  rods  ? 

Ans.,  227.5  square  rods. 

Ex.  2,  What  is  the  area  of  a  square  whose  side  is  315  7 
foot  ?  Ans.,  99666.49  square  feet. 

Ex.  3.  What  is  the  area  of  a  rectangular  board  whose  length 
is  15,25  feet,  and  breadth  15  inches  ? 

Ans.,  19.0625  square  feet. 

Ex.  4.  How  many  square  yards  are  there  in  the  four  sides 
of  a  room  which  is  18  feet  long,  15  feet  broad,  and  9  feet  high  1 

Am.,  66  square  yards. 

(82.)  If  the  sides  and  angles  of  a  parallelogram  are  given, 
the  perpendicular  height  may  be  found  by         D  c 

Trigonometry.  For  DE  is  one  side  of  a 
right-angled  triangle,  of  which  AD  is  the 
hypothenuse.  Hence, 

R  :  AD  :  :  sin.  A  :  DE  ; 

AD  X  sin.  A 


from  which 
Therefore,  the 


R 
ABxADXsin.  A 


R 
Hence  we  derive 

RULE  II. 

Multiply  together  two  adjacent  sides,  and  the  sine  of  tht 
included  angle. 

Ex.  1.  What  is  the  area  of  a  parallelogram  having  an  angl« 
of  58°,  and  the  including  sides  36  and  25.5  feet  ? 

Ans.  The  area = 36 X 25.5 X. 84805  (natural  sine  of  58°)=^ 
778.508  square  feet. 

The  computation  will  generally  be  most  conveniently  per- 
formed by  logarithms. 

Ex.  2.  What  is  the  area  of  a  rhombus,  each  of  whose  sides 
is  21  feet  3  inches,  and  each  of  the  acute  angles  53°  20'  ? 

Ans.,  362.209  feet. 

Ex.  3.  How  many  acres  are  contained  in  a  parallelogram 
one  of  whose  angles  is  30°,  and  the  including  sides  are  25.3*1 
and  10.4  chains?  A.ns.,  13  acres,  29.12  rods 


MENSURATION  or  SURFACES.        61 

PROBLEM  II. 
(83.)  To  find  the  area  of  a  triangle. 

RULE  I. 

Multiply  the  base  by  half  the  altitude. 

For  demonstration,  see  Geometry,  Prop.  6,  B.  IV 

Ex.  1.  How  many  square  yards  are  contained  in  a  iriaLgle 
whose  base  is  49  feet,  and  altitude  25£  feet  ? 

Ans.,  68.736. 

Ex.  2.  "What  is  the  area  of  a  triangle  whose  base  is  45  feet, 
and  altitude  27.5  feet  ?  Ans.,  618.75  square  feet. 

(84.)  When  two  sides  and  the  included  angle  are  given,  \ro 
may  use 

RULE  II. 

Multiply  half  the  product  of  two  sides  by  the,  sine  of  tfis, 
included  angle. 

The  reason  of  this  rule  is  obvious,  from  Art.  82,  since  a  tri- 
angle is  half  of  a  parallelogram,  having  the  same  base  and  at- 
titude. 

Ex.  1.  What  is  the  area  of  a  triangle  of  which  two  sides  are 
45  and  32  feet,  and  the  included  angle  46°  30'  ? 

Ans.  The  area=45x  16  X. 725374  (natural  sine  of  46°  30')=- 

522.269  feet. 

Ex.  2.  What  is  the  area  of  a  triangle  of  which  two  sides  are 
127  and  96  feet,  and  the  included  angle  67°  15'  ? 

Ans. 
(85.)  When  the  three  sides  are  known,  we  may  use 

RULE  III. 

From  half  the  sum  of  the  three  sides  subtract  each  side  sev- 
erally ;  multiply  together  the  half  sum  and  the  three  remain* 
ders,  and  extract  the  square  root  of  the  product. 

Demonstration. 

Let  «,  &,  c  denote  the  sides  of  the  tri- 
angle ABC  ;  then,  by  Geometry,  Prop.  12, 
B,  IV.,  we  have  BC2=AB2+AC2-2ABx 
U),  or  a'--=£2-j-r;2-2f:X  AD  ;  whence, 


TRIGONOMETRY 


But  CD2-  AC2  -AD2; 

honcc      (^=b,_( 


CD=- 


But  /At  area= 


The  quantity  under  the  radical  sign  being  the  difference  ol 
two  squares,  may  be  resolved  into  the  factors  2&c+(63-fca—  a*) 
and  2bc—  (tf+cn~—  a?)  ;  and  these,  in  the  same  manner,  may 
be  resolved  into        (b+c+a)x(b+c—  #), 
and  (a+b  —  c)x(a—  b+c). 

Hence,  if  we  put  S  equal  to  -        —  ,  we  shall  have 

& 


the  arf>a=  \/S(S-a)  (~S-&)  (S-c). 

Ex.  1.  "What  is  lite  area  of  a  triangle  whose  sides  are  125, 
173,  and  216  feet  ? 

Here  S-257,  S-6=84, 

S—  a=132,  S-c=:4l. 

Hence  the  area=  V257  X  132  X  84  X  41  =  10809  square  feet. 
Ex.  2.  How  many  acres  arc  contained  in  a  triangle  whose 
sides  are  49,  50.25,  and  25.69  chains? 

Ans.j  61  acres,  1  rood,  39.68  perches. 
Ex.  3.  "What  is  the  area  of  a  triangle  whose  sides  are  234, 
289,  and  345  feet? 

Ans. 

(86.)  In  an  equilateral  triangle,  one  of  whose  sides  is  a,  tho 
expression  for  the  area  becomes 


_oV3 

4     ; 

that  is,  the  area  of  an  equilateral  triangle  is  equal  to  on 
.ourth  the  square  of  one  of  its  sides  multiplied  by  the  square 
root  of  3. 

Ex.  What  is  the  area  of  a  triangle  whose  sides  are  each  37 
feet?  Ans.,  592.79  feet 


MENSURATION  OF  SURFACES.        6S 

PROBLEM  III. 
(87.)  To  find  the  area  of  a  trapezoid. 

RULE. 

Multiply  half  the  sum  of  the  parallel  sides  into  their  per 
ptndicular  distance. 

For  demonstration,  see  Geometry,  Prop.  7,  B.  IV. 

Ex.  1.  "What  is  the  area  of  a  trapezoid  whose  parallel  sidea 
are  156  and  124,  and  the  perpendicular  distance  between  them 
57  feet? 

Ans.,  7980  feet. 

Ex.  2.  How  many  square  yards  in  a  trapezoid  whose  par- 
allel sides  are  678  and  987  feet,  and  altitude  524  feet  ? 

Ans. 

PROBLEM  IV. 
(88.)  To  find  the  area  of  an  irregular  polygon. 

RULE. 

Draw  diagonals  dividing  the  polygon  into  triangles,  and 
find  the  sum  of  the  areas  of  these  triangles. 

Ex.  1.  What  is  the  area  of  a  quadrilateral,  one  of  whoso 
diagonals  is  126  feet,  and  the  two  perpendiculars  let  fall  upo» 
it  from  the  opposite  angles  are  74  and  28  feet  ? 

Ans.,  6426  feet. 

Ex.  2.  In  the  polygon  ABODE,  there 
are  given  EC=205,  EB=242,  AF=65, 
DGr  =  114,  and  DH=110,  to  find  the  area. 

Ans. 

(89.)  If  the  diagonals  of  a  quadrilateral 
are  given,  the  area  may  be  found  by  the 
following 

RULE. 

Multiply  half  the  product  of  the  diagonals  by  the  sine  of 
the  angle  at  their  intersection. 

Demonstration. 
The  sines  of  the  four  angles  at  E  are  all  equal  tc  each  other 


TRIGONOMETRY. 


since  fat  adjacent  angles  AED,  DEC  are  the  supplements  01 
each  other  (Art.  27).     But,  according  to 
Ihe  Rule,  Art.  84,  the  area  of 
the  Mangle  ABE=iAExBEXsine  E  ; 
"         "       AED=|-AExDEXsine  E; 
"         "       BEC=iBExECXsine  E; 
"         "       DEC=i-DExECXsineE. 
Therefore, 

the  area  of  ABCD=J(AE-fEC)X(BE  +  ED)Xsine  E 

=jACxBDXsine  E. 

Ex.  1.  If  the  diagonals  of  a  quadrilateral  are  34  and  56 
rods,  and  if  they  intersect  at  an  angle  of  67°,  what  is  the  area? 

Ans.,  876.32. 

Ex.  2.  If  the  diagonals  of  a  quadrilateral  are  75  and  49, 

what  is  the  area  ? 
Ans. 


and  the  angle  of  intersection  is  42°, 


PROBLEM  Y. 
(90.)  To  find  the  area  of  a  regular  polygon. 

RULE  I. 

Multiply  half  the  perimeter  by  the  perpendicular  let  fall 
from  the  center  on  one  of  the  sides. 
For  demonstration,  see  Geometry,  Prop.  7,  B.  VI. 
Ex.  1.  "What  is  the  area  of  a  regular  pentagon  whose  side 
is  25,  and  the  perpendicular  from  the  center  17.205  feet  ? 

Ans.,  1075.31  feet. 

Ex.  2.  What  is  the  area  of  a  regular  octagon  whose  side  is 
53,  and  the  perpendicular  63.977  ? 

Ans. 

(01.)  "When  the  perpendicular  is  not  given,  it  may  "be  con. 
puted  from  the  perimeter  and  number  of  sides.  If  we  divide 
360  degrees  by  the  number  of  sides  of  the 
polygon,  the  quotient  will  be  the  angle  ACB 
at  the  center,  subtended  by  one  of  the  sides.  /  c 

The  perpendicular  CD  bisects  the  side  AB, 
and  the  angle  ACB.  Then,  in  the  triangle 
ACD,  we  have  (Art.  42), 

R  :  AD  :  cot.  ACD  :  CD  ;  that  is, 


A 


HE.\S    A  ATI  ON   OF    SURFACES.  66 

Radius  is  to  half  of  one  of  the  s\des  of  the  polygon,  as  the 
cotangent  of  the  opposite  angle  is  to  *he  perpendicular  from 
t\s  center. 

Ex.  3.  Find  the  area  of  a  regular  hexagon  whose  side  is  32 
inches. 

The  angle  ACD  is  TV  of  360°=30°.     Then 
R  :  16  :  :  cot.  30°  :  27.7128= CD,  the  perpendicular ; 
and  the  area=27.7128x!6x6=2660.42S8. 

Ex.  4.  Find  the  area  of  a  regular  decagon  whose  side  is  4b 
feet.  Ans.,  16280.946. 

(92.)  In  this  manner  was  computed  the  following  tahle  of 
tha  areas  of  regular  polygons,  in  which  the  side  of  each  poly- 
g<-Ti  is  supposed  to  be  a  unit. 

TABLE  OF  REGULAR  POLYGONS. 

Names.  Sides.  Areas. 

Triangle,                    3  0.4330127. 

Square,                      4  1.0000000. 

Pentagon,                   5  1.7204774. 

Hexagon,                   6  2.5980762. 

Heptagon,                  7  3.6339124 

Octagon,                    8  4.8284271 

Nonagon,                    9  6.1818242. 

Decagon,                  10  7.6942088. 

Undecagon,              11  9.3656399. 

Dodecagon, .             12  11.1961524. 

By  the  aid  of  this  table  may  be  computed  the  area  of  any 
other  regular  polygon  having  not  more  than  twelve  sides.  For, 
since  the  areas  of  similar  polygons  are  as  the  squares  of  their 
homologous  sides,  we  derive 

RULE  II. 

Multiply  the  square  of  one  of  the  sides  of  the  polygon  by 
the  area  of  a  similar  polygon  whose  side  is  unity. 

Ex.  5.  What  is  the  area  of  a  regular  nonagon  whose  side 
is  63  ?  Ans.,  24535.66. 

Ex.  6.  What  is  the  area  of  a  regular  dodecagon  whose  side 
is  54  feet?  Ans.,  32647.98  feet. 

E 


6tt  TRIGONOMETRY. 

PROBLEM  VI 
(93.)  To  find  the  circumference  of  a  circle  from  its  diamete* 

RULE 

Multiply  the  diameter  by  3.14159. 

"For  the  demonstration  of  this  rule,  see  Geometry,  Prop.  13, 
Cor.  2,  B.VI. 

When  the  diameter  of  the  circle  is  small,  and  no  great  ac- 
curacy is  required,  it  may  be  sufficient  to  employ  the  value 
of  TT  to  only  4  or  5  decimal  places.  But  if  the  diameter  is 
large,  and  accuracy  is  required,  it  will  be  necessary  to  employ 
a  corresponding  number  of  decimal  places  of  TT.  The  value  of 
TT  to  ten  decimal  places  is  3.14159,26536, 

and  its  logarithm  is  0.497150. 

Ex.  1.  "What  is  the  circumference  of  a  circle  whose  diame- 
ter is  125  feet  ? 

Ans.,  392.7  feet. 

Ex.  2.  If  the  diameter  of  the  earth  is  7912  miles,  what  in 
its  circumference  ? 

Ans.,  24856.28  miles. 

Ex.  3.  If  the  diameter  of  the  earth's  orbit  is  189,761,000 
miles,  what  is  its  circumference  ? 

Ans.,  596,151,764  miles. 

To  obtain  this  answer,  the  value  of  TT  must  be  taken  to  at 
least  eight  decimal  places. 

PROBLEM  VII. 

(94.)  To  find  the  diameter  of  a  circle  from  its  circum- 
ference. 

RULE  I. 

Divide  the  circumference  by  3.14159. 

This  rule  is  an  obvious  consequence  from  the  preceding 
To  divide  by  a  number  is  the  same  as  to  multiply  by  its  re- 
ciprocal ;  and,  since  multiplication  is  more  easily  performed 
than  division,  it  is  generally  most  convenient  to  multiply  by 
the  reciprocal  of  TT,  which  is  0.3183099.  Hence  we  have 

RULE  II. 
Multiply  the  circumference  by  0.31831. 


MENSURATION   OF   SURFACES.  67 

Ex.  1.  What  is  the  diameter  of  a  circle  whose  circumference 
b  875  foot  ? 

Ans.,  278.52  feet 

Ex.  2.  If  tho  circumference  of  the  moon  is  6786  miles,  what 
is  its  diameter  ? 

Ans.,  2160  miles. 

Ex.  3.  If  the  circumference  of  the  moon's  orbit  is  1,492,987 
miles,  what  is  its  diameter  ? 

Ans.,  475,233  miles 

PROBLEM  VIII. 
(95.)  To  find  the  length  of  an  arc  of  a  circle. 

RULE  I. 

As  360  is  to  the  number  of  degrees  in  the  arc,  so  is  the  cu- 
cumference  of  the  circle  to  the  length  of  the  arc. 

This  rule  follows  from  Prop.  14,  B.  III.,  in  Geometry,  where 
it  is  proved  that  angles  at  the  center  of  a  circle  have  the  same 
ratio  with  the  intercepted  arcs. 

Ex.  1.  "What  is  the  length  of  an  arc  of  22°,  in  a  circle  whose 
diameter  is  125  feet? 

The  circumference  of  the  circle  is  found  to  be  392.7  feet. 

Then  360  :  22  : :  392.7  :  23.998  feet. 
Ex.  2.  If  the  circumference  of  the  earth  is  24,856.28  miles 
what  is  the  length  of  one  degree  ? 

Ans.,  69.045  miles. 

RULE  II. 

(96.)  Multiply  the  diameter  of  the  circle  by  the  number  of 
degrees  in  the  arc,  and  this  product  by  0.0087266. 

Since  the  circumference  of  a  circle  whose  diameter  is  unity 
is  3.14159,  if  we  divide  this  number  by  360,  we  shall  obtain 
the  length  of  an  arc  of  one  degree,  viz.,  0.0087266.  If  we 
multiply  this  decimal  by  the  number  of  degrees  in  any  arc,  we 
shall  obtain  the  length  of  that  arc  in  a  circle  whose  diameter 
is  unity  ;  and  this  product,  multiplied  by  the  diameter  of  any 
other  circle,  will  give  the  length  of  an  arc  of  the  given  num- 
ber of  degrees  in  that  circle. 


tiS  TRIGONOMETRY. 

Ex.  3.  What  is  the  length  of  an  arc  of  25°,  in  a  circle  whose 
radius  is  44  rods  ? 

Ans.,  19.198  rods. 

Ex.  4.  What  is  the  length  of  an  arc  of  11°  15',  in  a  circle 
whose  diameter  is  1234  feet  ? 

Ans.,  121.147  feet. 

(97.)  If  the  number  of  degrees  in  an  arc  is  not  given,  it  m?  ? 
be  computed  from  the  radius  of  the  circle, 
and  either  the  chord  or  height  of  the  arc. 
Thus,  let  AB  be  the  chord,  and  DE  the 
height  of  the  arc  ADB,  and  C  the  center 
of  the  circle.  Then,  in  the  right-angled  tri- 
angle ACE, 

AC  •  R  •  •  I  AE  :  sin*  ACE) 
:.(CE  :  cos.  ACE, 

either  of  which  proportions  will  give  the  number  of  degrees  in 
half  the  arc.  . 

If  only  the  chord  and  height  of  the  arc  are  given,  the  diam- 
eter of  the  circle  may  be  found.  For,  by  Geometry,  Prop.  22, 
Cor.,  B.  IV., 

DE  :  AE  : :  AE  :  EF. 

Ex.  5.  What  is  the  length  of  an  arc  whose  chord  is  6  feet, 
in  a  circle  whoso  radius  is  9  feet  ? 

Ans.,  6.117  feet. 

PROBLEM  IX. 
(98.)  To  find  the  area  of  a  circle. 

RULE  I. 

Multiply  the  circumference  by  half  the  radius. 
For  demonstration,  see  Geometry,  Prop.  12,  B.  VI. 

RULE  II. 

Multiply  t/ie  square  of  the  radius  by  3.14159, 
See  G-eometry,  Prop  13,  Cor.  3,  B.  VI. 
Ex,  1.  What  is  the  area  of  a  circle  whose  diameter  is  iH 
feet? 

Ans.,  254.469  feet. 


MENSURATION   OF   SURFACES  fi8 

Ex  2.  What  is  the  area  of  a  circle  whose  circumference  is 
74  feet? 

Ans.,  435.766  feet. 

Ex.  3.  What  is  the  area  of  a  circle  whose  radius  is  125 
yards  ? 

Ans.,  49087.38  yards 

PROBLEM  X. 
(99.)  To  find  the  area  of  a  sector  of  a  circle. 

RULE  I 

Multiply  the  arc  of  the  sector  by  half  its  radius. 
See  Geometry,  Prop.  12,  Cor.,  B.  VI. 

RULE  II. 

As  360  is  to  the  number  of  degrees  in  the  arc,  so  is  the  area 
nf  the  circle  to  the  area  of  the  sector. 

This  follows  from  Geometry,  Prop.  14,  Cor.  2,  B.  III. 
Ex.  1.  What  is  the  area  of  a  sector  whose  arc  is  22°,  in  a 
circle  whose  diameter  is  125  feet  ? 

The  length  of  the  arc  is  found  to  be  23.998. 
Hence  the  area  of  the  sector  is  749.937. 
Ex.  2.  What  is  the  area  of  a  sector  whose  arc  is  25°,  in  a 
circle  whose  radius  is  44  rods  ? 

Ans.,  422.367  rods. 

Ex.  3.  What  is  the  area  of  a  sector  less  than  a  semicircle, 
whose  chord  is  6  feet,  in  a  circle  whose  radius  is  9  feet  ? 

Ans.,  27.522  feet 

PROBLEM  XI. 
(100.)  To  find  the  area  of  a  segment  of  a  circle. 

RULE. 

Find  the  area  of  the  sector  which  has  the  same  arc,  and 
also  the  area  of  the  triangle  formed  by  Ike  chord  of  the  seg- 
ment and  the  radii  of  the  sector. 

Then  take  the  sum  of  these  areas  if  the  segment  is  greater 
than  a  semicircle,  but  take  their  difference  if  it  is  less 


fO  TRIGONOMETRY 

It  is  obvious  that  the  segment  AEB  is 
equal  to  the  sum  of  the  sector  ACBE  and 
the  triangle  ACB,  and  that  the  segment 
ADB  is  equal  to  the  difference  "between 
the  sector  ACBD  and  the  triangle  ACB. 

Ex.  1.  What  is  the  area  of  a  segment 
whose  arc  contains  280°,  in  a  circle  whoso 
diameter  is  50  ? 

The  whole  circle       =       1963.495 
The  sector  =       1527.163 

The  triangle  307.752 

The  segment  =       1834.915,  Ans. 

Ex.  2.  What  is  the  area  of  a  segment  whose  chord  is  20  Aeet, 
and  height  2  feet  ? 

Ans.,  26.8788  feet. 

Ex.  3.  What  is  the  area  of  a  segment  whose  arc  is  25°,  in 
a  circle  whose  radius  is  44  rods  ? 

Ans. 

(101.)  The  area  of  the  zone  ABHGr,  included  between  two 
parallel  chords,  is  equal  to  the  difference  between  the  segments 
GDI!  and  ADB. 

Ex.  4.  What  is  the  area  of  a  zone,  one  side  of  which  is  96, 
and  the  other  side  60,  and  the  distance  between  them  26  ? 

Ans.,  2136.7527. 

The  radius  of  the  circle  in  this  example  will  be  found  to 
be  50. 

PROBLEM  XII. 

(102.)  To  find  the  area  of  a  ring-  included  between  the  cir 
eumferences  of  two  concentric  circles. 

RULE. 

Take  the  difference  betiveen  the  areas  of  the  two  circles;  or, 
Subtract  the  square  of  the  less  radius  from  the  square  of  the 
greater,  and  multiply  their  difference  by  3.14159. 
For,  according  to  Geometry,  Prop.  13,  Cor.  3,  B.  VI., 
the  area  of  the  greater  circle  is  equal  to  TT  Ra, 
and  the  area  of  the  smaller,  TT  r*. 

Their  difference,  or  the  area  of  the  ring,  b  IT  (R3— y2). 


MENSURATION   OF    SOLUS.  7. 

Ex.  1.  The  diameters  of  two  concentric  circles  are  60  and 
50.  What  is  the  area  of  the  ring  included  "between  their  cir- 
cumferences ? 

Ans.,  863.938. 

Ex.  2.  The  diameters  of  two  concentric  circles  are  320  and 
280  What  is  the  area  of  the  ring  included  between  their  cir- 
cumferences ? 

Ans.,  18849.55 

PROBLEM  XIII. 
(103.)  To  find  the  area  of  an  ellipse. 

RULE. 

Multiply  the  product  of  the  semi-axes  by  3.14159. 
For  demonstration,  see  Greometry,  Ellipse,  Prop.  21. 
Ex.  1.  What  is  the  area  of  an  ellipse  whose  major  axis  i* 
70  feet,  and  minor  axis  60  feet  ? 

Ans.,  3298.67  feet. 

Ex.  2.  What  is  the  area  of  an  ellipse  whose  axes  are  340 
and  310? 

Ans.,  82780.896 

PROBLEM  XIY. 
(104.)  To  find  the  area  of  a  parabola. 

RULE. 

Multiply  the  base  by  two  thirds  of  the  height. 
For  demonstration,  see  Greometry,  Parabola,  Prop.  12. 
Ex.  1.  What  is  the  area  of  a  parabola  whose  base  is  18  feet, 
and  height  5  feet  ? 

Ans.,  60  feet. 

Ex.  2.  What  is  the  area  of  a  parabola  whose  base  is  525 
feet,  and  height  350  feet  ? 

Ans.,  122500  feet 

MENSURATION  UF  SOLIDS. 

(105.)  The  common  measuring  unit  of  solids  is  a  cubct 
•vhose  faces  are  squares  of  the  same  name ;  as,  a  cubic  inch 
a  cubic  foot,  &c.  This  measuring  unit  is  not,  however,  of 


72  TRIGONOMETR* 

necessity  a  cube  whose  faces  are  squares  of  the  same  name*. 
Thus  a  bushel  may  have  the  form  of  a  cube,  but  its  faces  can 
only  be  expressed  by  means  of  some  unit  of  a  different  denom- 
ination. The  following  is 

The  Table  of  Solid  Measure. 

1728     cubic  inches  =  1  cubic  foot. 

27     cubic  feet       =  1  cubic  yard. 

4492J  cubic  feet       =  1  cubic  rod. 

231     cubic  inches  =  1  gallon  (liquid  measure). 

268.8  cubic  inches  =  1  gallon  (dry  measure). 

2150.4  cubic  inches  =  1  bushel. 

PROBLEM  I. 

(106.)  To  find  the  surface  of  a  right  prism. 

•  .  • 
RULE. 

Multiply  the  perimeter  of  the  base  by  the  altitude  for  the 
convex  surface.     To  this  add  the  areas  of  the  two  ends  when 
the  entire  surface  is  required. 
See  G-eometry,  Prop.  1,  B.  VIII. 

Ex.  1.  What  is  the  entire  surface  of  a  parallelepiped  whose 
altitude  is  20  feet,  breadth  4  feet,  and  depth  2  feet  ? 

Ans.,  256  square  feet. 

Ex.  2.  What  is  the  entire  surface  of  a  pentagonal  prism 
whose  altitude  is  25  feet  6  inches,  and  each  side  of  its  base  3 
feet  9  inches  ? 

Ans.j  526.513  square  feet. 

Ex.  3.  What  is  the  entire  surface  of  an  octagonal  prism 
whose  altitude  is  12  feet  9  inches,  and  each  side  of  its  base  1? 
feet  5  inches  ? 

Ans.,  302.898  square  f  jot. 

PROBLEM  II. 
(107.)  To  find  the  solidity  of  a  pnsm. 

iiULE. 

Multiply  the  area  of  the  base  by  the  altitude. 
See  Geometry,  Prop.  11,  B.  VIII. 


MENSURATION   OF   SOLIDS  7C 

Ex.  1.  What  is  the  solidity  of  a  parallelepiped  whose  alii, 
tude  is  30  feet,  breadth  6  feet,  and  depth  4  feet  ? 

Am.,  720  cubic  feet. 

Ex.  2..  What  is  the  solidity  of  a  square  prism  whose  altitude 
is  8  feet  10  inches,  and  each  side  of  its  base  2  feet  3  inches  ? 

Am.,  44 1 f  cubic  feet. 

Ex.  3  What  is  the  solidity  of  a  pentagonal  prism  whoso  a.« 
titude  is  20  feet  6  inches,  and  its  side  2  feet  7  inches  ? 

Ans.)  235.376  cubic  feet. 

PROBLEM  III. 
(108.)  To  find  the  surface  of  a  regular  pyramid. 

RULE. 

Multiply  the  perimeter  of  the  base  by  half  the  slant  height 
for  the  convex  surface.     To  this  add  the  area  of  the  bast 
when  the  entire  surface  is  required. 
See  Geometry,  Prop.  14,  B.  VIII. 

Ex.  1.  What  is  the  entire  surface  of  a  triangular  pyramid 
whose  slant  height  is  25  feet,  and  each  side  of  its  base  5  feet? 

Ans.^  198.325  square  feet. 

Ex.  2.  What  is  the  entire  surface  of  a  square  pyramid 
whose  slant  height  is  30  feet,  and  each  side  of  the  base  4  foot'.' 

Ans.)  256  square  fe'jt. 

Ex.  3.  What  is  the  entire  surface  of  a  pentagonal  pyra- 
mid \vhose  slant  height  is  20  feet,  and  each  side  of  the  base 
8  feet? 

Ans.)  165.484  square  feet. 

PROBLEM  IY. 
(109.)  To  find  the  solidity  of  a  pyramid.    . 

RULE. 

Multiply  the  area  of  the  base  by  one  third  of  the  altitude. 
See  Geometry,  Prop.  17,  B.  VIII. 

Ex.  1.  What  is  the  solidity  of  a  triangular  pyramid  whu*« 
altitude  is  25  feet,  and  each  sid  3  of  its  base  6  feet  ? 

Ans.,  129,904  cubic  feet 


7  4  TRIGONOMETRY. 

Ex.  2.  What  is  the  solidity  of  a  square  pyramid  whose  slant 
height  is  22  feet,  and  each  side  of  its  "base  10  feet  ? 

Ans.,  714.143  cubic  feet. 

Ex.  3.  What  is  the  solidity  of  a  pentagonal  pyramid  whosw 
altitude  is  20  feet,  and  each  side  of  its  base  3  feet  ? 

Ans.,  103.228  cubic  feet. 

PROBLEM  V. 

(110.)  To  find  the  surface  of  a  frustum  of  a  regular  pyr- 
amid. 

RULE. 

Multiply  half  the  slant  height  by  the  sum  of  the  perime- 
ters of  the  two  bases  for  the  convex  surface.  To  this  add 
the  areas  of  the  two  bases  when  the  entire  surface  is  re- 
quired. 

See  Geometry,  Prop.  14,  Cor.  1,  B.  VIII. 

Ex.  1.  What  is  the  entire  surface  of  a  frustum  of  a  square 
pyramid  whose  slant  height  is  15  feet,  each  side  of  the  greater" 
base  being  4  feet  6  inches,  and  each  side  of  the  less  base  2  feet 
10  inches  ? 

Ans.9  248.278  square  feet. 

Ex.  2.  What  is  the  entire  surface  of  a  frustum  of  an  oc- 
tagonal pyramid  whose  slant  height  is  14  feet,  and  the  sides 
of  the  ends  3  feet  9  inches,  and  2  feet  3  inches  ? 

Ans.,  428.344  square  feet. 

PROBLEM  VI. 
(111.)  To  find  the  solidity  of  a  frustum  of  a  pyramid. 

RULE. 

Add  together  the  areas  of  the  two  bases ,  and  a  mean  pro- 
portional between  them,  and  multiply  the  sum  by  one  thira 
of  the  altitude. 

See  Geometry,  Prop.  18,  B.  VIII. 

When  the  pyramid  is  regular,  it  is  generally  most  conven- 
ient to  find  the  area  of  its  base  by  Rule  II.,  Art.  92.  If  we 
put  a  to  represent  one  side  of  the  lower  base,  and  b  one  side 
of  the  upper  base,  and  *.he  tabular  number  from  Art.  92  b> 


MENSURATION   c  F    SOLIDS.  72 

T,  the  area  of  the  lower  base  will  be  a2T  ;  that  of  the  upper 
base  will  be  &2T  ;  and  the  mean  proportional  will  bo  a&T. 
Hence,  if  we  represent  the  height  of  the  frustum  -by  /*,  its  so- 
lidity will  be 


Ex.  1.  "What  is  the  solidity  of  a  frustum  of  an  hexagonal 
pyramid  whose  altitude  is  15  feet,  each  side  of  the  greater  end 
being  3  feet,  and  that  of  the  less  end  2  feet  ? 

Ans.,  246.817  cubic  feet. 

Ex.  2.  What  is  the  solidity  of  a  frustum  of  an  octagonal 
pyramid  whose  altitude  is  9  feet,  each,  side  of  the  greater  end 
being  30  inches,  and  that  of  the  less  end  20  inches  ? 

Ans.,  191.125  cubic  feet 

Definition. 

(112.)  A  wedge  is  a  solid  bounded  by  five  planes,  viz.,  a  rec- 
tangular base,  ABCD,  two  trape- 
zoids,  ABFE,  DCFE,  meeting  in 
an  edge,  and  two  triangular  ends, 
ADE,  BCF.  The  altitude  of  the 
wedge  is  the  perpendicular  drawn 
from  any  point  in  the  edge  to  the 
plane  of  the  base,  as  EH. 

PROBLEM  VII. 
(113.)  To  find  the  solidity  of  a  wedge. 

RULE. 

Add  the  length  of  the  edge  to  twice  the  length  of  the  base, 
and  multiply  the  sum  by  one  sixth  of  the  product  of  the  height 
of  the  ivedge  and  the  breadth  of  the  base. 

Demonstration. 

Put  L=AB,  the  length  of  the  baso  ; 
"     /=EF,  the  length  of  the  edge  ; 
"     £=BC,  the  breadth  of  the  base  ; 
"     /i=EH,  the  altitude  of  the  wedge. 
Now,  if  the  length  of  the  base  is  equal  to  that  of  the  e 


f(5  TRIGONOMETRY. 

it  is  evident  that  the  wedge  is  half  of  a  pri&m  of  the  same  base 
and  height.  If  the  length  of  the  base  is  greater  than  that  of 
the  edge,  let  a  plane,  EGrI,  be  drawn  parallel  to  BCF.  The 
wedge  will  le  divided  into  two  parts,  viz.,  the  pyramid  E- 
AIGrD,  and  the  triangular  prism  BCF—  Or. 

The  solidity  of  the  former  is  equal  to  £M(L  —  /),  and  that 
of  the  latter  is  \blil.     Their  sum  is 


If  the  length  of  the  base  is  less  than  that  of  the  edge,  the 
wedge  will  be  equal  to  the  difference  between  the  prism 
pyramid,  and  we  shall  have 

which  is  equal  to 

the  same  result  as  before. 

Ex.  1.  What  is  the  solidity  of  a  wedge  whose  base  is  3C 
inches  long  and  5  inches  broad,  its  altitude  12  inches,  and  the 
length  of  the  edge  2  feet  ? 

Ans.y  840  cubic  inches. 

Ex.  2.  What  is  the  solidity  of  a  wedge  whose  base  is  40 
inches  long  and  7  inches  broad,  its  altitude  18  inches,  and  ths 
length  of  the  edge  30  inches  ? 

Ans.,  2310  cubic  inches. 

Definition. 

(114.)  A  rectangular  prismoid  is  a  solid  bounded  by  six 
planes,  of  which  the  two  bases  are  rectangles  having  their  cor- 
responding  sides  parallel,  and  the  four  upright  sides  of  the  sol- 
id are  trapezoids. 

PROBLEM  VIII. 
To  find  the  solidity  of  a  rectangular  prismoid. 

RULE. 

Add  together  the  areas  of  the  two  bases,  and  four  times  tnt 
area  of  a  parallel  section  equally  distant  from  the  bases,  and 
multiply  the  sum  by  one  sixth  of  the  altitude. 

Demonstration. 
Put  L  and  B=  length  and  breadth  of  one  baso  ; 


MENSURATION   OF    SOLIDS.  77 

Put  /  and  b  =  length  and  breadth  of  the  other  base; 
«   M    "   m=  length  and  breadth  of  middle  sec.;    ^—  -  ~~ 
"    h  =the  altitude  of  the  prismoid.  --£     '»  >» 

It  is  evident  that  if  a  plane  be  made  to  pass 

through  the  opposite  edges  of  the  upper  and 

lower  bases,  the  prismoid  will  be  divided  into 

two  wedges,  whose  bases  are  the  bases  of  the 

prismoid,  and  whose  edges  are  L  and  L     The  solidity  of  these 

wedges,  and,  consequently,  that  of  the  prismoid,  is 


But,  since  M  is  equally  distant  from  L  and  /,  we  have 

2M=L+Z,  and  2m=~B+b  ; 
hence         4Mw=(L+/)  (B+6)=BL+B/+6L+W. 

Substituting  4Mm  for  its  value  in  the  preceding  expression, 
we  obtain  for  the  solidity  of  the  prismoid 


Ex.  1.  "What  are  the  contents  of  a  log  of  wood,  in  the  form 
of  a  rectangular  prismoid,  the  length  and  breadth  of  one  end 
oeing  16  inches  and  12  inches,  and  of  the  other  7  inches  and 
4  inches,  the  length  of  the  log  being  24  feet  ? 

Ans.,  16  J-  cubic  feet. 

Ex.  2.  What  is  the  solidity  of  a  log  of  hewn  timber,  whoso 
nids  are  18  inches  by  15,  and  14  inches  by  11  J,  its  length  be- 
ing 18  feet?  Ans.,  26fi  cubic  feet. 

PROBLEM  IX. 

To  compute  the  excavation  or  embankment  for  a  rail-way 
(115.)  By  the  preceding  rule  may  be  computed  the  amount 
if  excavation  or  embankment  required  in  constructing  a  rail- 
«oad  or  canal.  If  we  divide  the  line  of  the  road  into  portions 
jo  small  that  each  may  be  regarded  as  a  straight  line,'  and 
suppose  an  equal  number  of  transverse  sections  to  be  made, 
the  excavation  or  embankment  between  two  sections  may  be 
regarded  as  a  prismoid,  and  its  contents  found  by  the  pre 
ceding  rule. 

Let  ABCD  represent  the  lower  surface  of  the  supposed  ex- 
oavation,  which  we  will  assume  to  be  parallel  to  the  horizon  ; 
tind  let  EFG-H  represent  the  upper  surface  of  the  excavation 


TRIGONOMETRY. 


projected  on  a  horizontal  plane.  Also,  let  E'A'B'F',  G'C'I)  H 
represent  the  vertical  sections  at 
the  extremities.  If  we  suppose  ver- 
tical planes  to  pass  through  the  lines 
AC,  BD,  the  middle  part  of  the  ex- 
cavation, or  that  contained  between 
these  vertical  planes,  will  be  a  rect- 
angular prismoid,  of  which  A'B'KI 
will  be  one  base,  and  C'D'ML  the 
other  base.  Its  solidity  will  there- 
fore be  given  by  Art.  114.  The 
parts  upon  each  side  of  the  middle  prismoid  are  also  halves  of 
rectangular  prismoids  ;  or,  if  the  two  parts  are  equal,  they 
may  be  regarded  as  constituting  a  second  prismoid,  one  of 
whose  bases  is  the  sum  of  the  triangles  A'E'I,  B'F'K  ;  and  the 
other  base  is  the  sum  of  the  triangles  C'Gr'L,  D'H'M.  There- 
fore the  volume  of  the  entire  solid  is  equal  to  the  product  of 
one  sixth  of  its  length,  by  the  sum  of  the  areas  of  the  sections 
at  the  two  extremities,  and  four  times  the  area  of  a  parallel 
and  equidistant  section. 

Ex.  1.  Let  ABCDEFGr  represent  the  profile  of  a  tiact  of 


land  selected  for  the  line  of  a  rail- way ;  and  suppose  it  is  re- 
quired, by  cutting  and  embankment,  to  reduce  it  from  its  pres- 
ent hilly  surface  to  one  uniform  slope  from  the  point  A  to 
the  point  GK 

The  distance  AH  is  561  feet ;  the  distance  DK  is  820  feet; 

"          "       HI  is  858  feet;     "         "       KL  is  825  feet; 

"          "       ID    is  825  feet ;     "         "       LG-  is  330  feet. 

The  perpendicular  BH  is  18  feet ;  the  perpendicular  KE  is  19  feel 

"  «  CI  is20feet;    "  "  LF  in   8 feet 

The  annexed  figure   repre- 
sents a  cross  section,  showing 
the  form  of  the  excavation. 
The  base  of  the  cutting  is  to 


MENSURATION   OF    Sc  LIDS.  7$ 

be  50  feet  wide,  the  slope  1J  horizontal  to  1  perpendicular; 
that  is,  where  the  depth  ad  is  10  feet,  the  width  of  the  slope 
c.d  at  the  surface  will  be  15  feet. 

Calculation  of  the  portion  ABH. 

Since  BH  is  18  feet,  the  length  of  cd  in  the  cross  section 
will  be  27  feet,  and  c/,  the  breadth  at  the  top  of  the  section, 
will  be  104  feet.  We  accordingly  find,  by  Art.  87,  the  area  of 
the  trapezoid  forming  the  cross  section  at  BH  equal  to 

1^X18=1386  fat 

For  the  middle  section,  the  height  is  3  feet,  cd  is  13.5  feet, 
and  cf  is  77  feet.  The  area  of  the  cross  section  is  therefore 
equal  to 


The  solid  ABH  will  therefore  be  equal  to 

fr*-\ 

(1386+4x571.5)  -^-=343332  cubic  feet,  or 
12716  cubic  yards. 

Calculation  of  the  portion  BCIH. 

Since  CI  is  20  feet,  the  length  of  cd  is  30  feet,  and  cf  is  1  (0 
feet.     The  area  of  the  section  at  CI  is  therefore  equal  to 


For  the  middle  section,  the  height  is  19  feet,  cd  is  28.5  foet, 
and  cf  is  107  feet.  The  area  of  the  cross  section  is  therefore 
equal  to 

1^X19=  1491.5. 

« 

The  solid  BCIH  will  therefore  be  equal  to 

oro 

(1386+1600+4x1491.5)  -g-=  1280136  cubic  feet,  or 
47412.4  cubic  yards. 

Calculation  of  the  portion  CID. 

The  height  of  the  middle  section  is  10  feet  ;  therefore  cf  is 
80  feet,  and  the  area,  of  the  cross  section  is 


TRIGONOMETRY. 


The  solid  C1D  will  therefore  be  equal  to 

DQ  K 

(1600+4X650)  -^-=577500  cubic  fcetj  02 

21388.9  cubic  yards. 

The  entire  amount  of  excavation  therefore  is, 
ABH=  12716.0  cubic  yards. 
BCIH=47412.4          " 
CDI=21388.9  " 

Total  excavation,    81517.3  " 


The  following  is  a  cross  section,  showing  the  ftrm  of  the  era 
bankment. 

The  top  of  the  embankment  ^ j 

is  to  be  50  feet  wide,  the  slope 
2  to  1 ;  that  is,  where  the  height 
ad  is  10  feet,  the  base  cd  is  to 
be  20  feet. 

Calculation  of  the  portion  DKE. 

Since  EK  is  19  feet,  the  length  of  cd  is  38  feet,  and  cf  is 
126  feet.  The  area  of  the  cross  section  at  EK  is  therefore 
equal  to 

1^X19=1672. 

# 

For  the  middle  section,  the  height  is  9.5  feet ;  cd  is  therefore 
19  feet,  and  cf  is  88  feet.  The  area  of  the  cross  section  is 
therefore 

88+50 


The  solid  DKE  is  therefore  equal  to 

820 
(1672+4  X  655.5)  -g-= 586846.7  cubic  feet,  or 

21735.1  cubic  yards! 

Calculation  of  the  portion  KEFL. 
Since  LF  is  8  feet,  cd  is  16  feet,  and  cf  is  82  feet, 
area  of  the  section  at  LF  is  therefore  equal  to 


MENSURATION   V.F    SOLIDS.  81 


The  height  of  the  middle  section  is  13.5  feei;  therefore  cd 
b  27  feet,  and  cf  is  104  feet.  The  area  of  the  cross:  section  is 
therefore  equal  to 

1^x13.5=1039,5. 

The  solid  KEFL  will  therefore  be  equal  to 


(1672+528+4xl039.5)-g-=  874225  cubic  feet,  or 
32378.7  cubic  yards. 

Calculation  of  tke  portion  LFGr. 

The  height  of  the  middle  section  is  4  feet  ;  therefore  cf  is  66 
*eet,  and  the  area  of  the  cross  section  is  equal  to 
66+50 


The  solid  LFGr  will  therefore  be  equal  to 

OOQ 

(528  +4  X  232)  -g-=  80080  cubic  feet,  or 

2965.9  cubic  yards. 
The  entire  amount  of  embankment  therefore  is 

DKE  =21735.1  cubic  yards. 
KEFL=32378.7  " 

LFG-=  2965.9 

Total  embankment,  57079.7  " 

Ex.  2.  Compute  the  amount  of  excavation  of  the  hill  ABCD 
from,  the  following  data  : 

The  distance  AH  is  325  feet  ;  the  perpendicular  BH  is  12  feet  ; 
"  "       HI  is  672  feet;     "  "  CI  is  13  feet, 

"  "       ID  is  534  feet. 

The  base  of  the  cutting  to  be  50  feet  wide,  and  the  slope  1^ 
horizontal  to  1  perpendicular.          Am.,  33969  cubic  yards 

PROBLEM  X. 
(116.)  To  find  the  surface  of  a  regular  polyedron. 

RULE. 

Multiply  the  area  of  one  of  the  faces  by  the  number  oj 

V 


S2j  TRIGONOMETRY. 

faces;  or,  Multiply  the  square  of  one  of  the  edges  by  tht 
surface  of  a  similar  solid  whose  edge  is  unity. 

Since  all  the  faces  of  a  regular  polyedron  are  equal,  it  is 
evident  that  the  area  of  one  of  them,  multiplied  by  their  num- 
ber, will  give  the  entire  surface.  Also,  regular  solids  of  the 
same  name  are  similar,  and  similar  polygons  are  as  the  squares 
of  their  homologous  sides  (Geom.,  Prop.  26,  B.  IV.).  The  fol- 
lowing table  shows  the  surface  and  solidity  of  regular  poly- 
edrons  whose  edge  is  unity.  The  surface  is  obtained  by  mul- 
tiplying the  area  of  one  of  the  faces,  as  given  in  Art.  92,  by 
the  number  of  faces.  Thus  the  area  of  an  equilateral  trian- 
gle, whose  side  is  1,  is  0.4330127.  Hence  the  surface  of  a 
regular  tetraedron 

=  .4330127  X  4= 1.7320508, 
and  so  on  for  the  other  solids. 

A  Table  of  the  regular  Polyedrons  whose  Edges  are  unity 


Names. 

No.  of  Faces. 

Surface. 

Solidity. 

Tetraedron, 

4 

1.7320508 

0.1178513. 

Hexaedrpn, 

6 

6.0000000 

1.0000000. 

Octaedron, 

8 

3.4641016 

0.4714045 

Dodecaedron, 

12 

20.6457288 

7.6631189. 

Icosaedron, 

20 

8.6602540 

2.1816950. 

Ex.  1.  What  is  the  surface  of  a  regular  octaedron  whose 
rilges  are  each  8  feet? 

Ans.,  221.7025  feet. 

Ex.  2.  What  is  the  surface  of  a  regular  dodecaedron  whose 
se  is  12  feet  ? 

Ans.,  2972.985  feet 

PROBLEM  XI. 
(117.)   To  find  the  solidity  of  a  regular  polyedron. 

RULE. 

3Iulliply  the  surface  by  one  third  of  the  perpendicular  let 
from  the  center  on  one  of  the  faces:  or,  Multiply  the 
i.ube  of  one  of  the  edges  by  the  solidity  of  a  similar  polyedron, 
whose  edgt  is  unity. 

Since  the  *•  ices  of  a  r.xnilar  polyedron  are  similar  and  equal. 


MENSURATION   OF    SOLID s.  83 

and  the  solid  angles  are  al]  equal  to  each  other,  it  is  evident 
that  the  faces  are  all  equally  distant  from  a  point  in  the  solid 
called  the  center.  If  planes  be  made  to  pass  through  the  cen- 
cer  and  the  several  edges  of  the  solid,  they  will  divide  it  into 
as  many  equal  pyramids  as  it  has  faces.  The  base  of  each 
pyramid  will  be  one  of  the  faces  of  the  polyedron ;  and  since 
their  altitude  is  the  perpendicular  from  the  center  upon  one  of 
the  faces,  the  solidity  of  the  polyedron  must  be  equal  to  the 
areas  of  all  the  faces,  multiplied  by  one  third  of  this  perpen- 
dicular. 

Also,  similar  pyramids  are  to  each  other  as  the  cubes  ol 
their  homologous  edges  (Geom.,  Prop.  17,  Cor.  3,  B.  VIIL). 
And  since  two  regular  polyedrons  of  the  same  name  may  be 
divided  into  the  same  number  of  similar  pyramids,  they  must 
be  to  each  other  as  the  cubes  of  their  edges, 

(118.)  The  solidity  of  a  tetraedron  whose  edge  is  unity,  may 
ne  computed  in  the  following  manner  : 

Let  C— ABD  be  a  tetraedron.  From  one  angle,  C,  let  fal1. 
a  perpendicular,  CE,  on  the  opposite  face;  c 

draw  EF  perpendicular  to  AD;  and  join 
CF,  AE.  Then  AEF  is  a  right-angled  tri- 
angle, in  which  EF,  being  the  sine  of  30°, 
is  one  half  of  AE  or  BE  ;  and  therefore 
FE  is  one  third  of  BF  or  CF.  Hence  the 
cosine  of  the  angle  CFE  is  equal  to  J ;  that  A. 
is,  the  angle  of  inclination  of  the  faces  of  the  polyedron  is  70° 
31'  44".  Also,  in  the  triangle  CAF,  CF  is  the"  sine  of  60°, 
which  is  0.866025.  Hence,  in  the  right-angled  triangle  CEF, 
knowing  one  side  and  the  angles,  we  can  compute  CE,  which 
is  found  to  be  0.8164966.  Whence,  knowing  the  base  ABD 
(Art.  92),  we  obtain  the  solidity  of  the  tetraedron  —0.1178513. 

In  a  somewhat  similar  manner  may  the  solidities  of  the 
other  regular  polyedrons,  given  in  Art.  116,  be  obtained, 

Ex.  1.  What  is  the  solidity  of  a  regular  tetraedron  whoso 
edges  are  each  24  inches  ? 

Ans.,  0.9428  feet. 

Ex.  2.  "What  is  the  solidity  of  a  regular  icosaedron  whoso 
ylges  are  each  20  feet  ? 

Ans.   17453.56  fret 


#4  TRIGONOMETRY. 

THE  THREE  ROUND  BODIES. 

PROBLEM  I. 
(119.)  To  find  the  surface  of  a  cylinder. 

RULE. 

Multiply  the  circumference  of  the  base  by  the  altitude  fm 
the  convex  surface.     To  this  add  the  areas  of  the  two  endi 
when  the  entire  surface  is  required. 
See  G-eometry,  Prop.  1,  B.  X. 

Ex.  1.  What  is  the  convex  surface  of  a  cylinder  whose  alti- 
tude is  23  feet,  and  the  diameter  of  its  base  3  feet? 

Ans.,  216.77  square  feet. 

Ex.  2.  What  is  the  entire  surface  of  a  cylinder  whoso  alti« 
tude  is  18  feet,  and  the  diameter  of  its  base  5  feet  ? 

Ans. 

PROBLEM  II. 
(120.)  To  find  the  solidity  of  a  cylinder. 

RULE. 

Multiply  the  area  of  the  base  by  the  altitude. 
See  Geometry,  Prop.  2,  B.  X. 

Ex.  1.  What  is  the  solidity  of  a  cylinder  whose  altitude  i? 
IS  feet  4  inches,  and  the  diameter  of  its  base  2  feet  10  inches? 

Ans.,  115.5917  cubic  feet. 

Ex.  2.  What  is  the  solidity  of  a  cylinder  whose  altitude  is 
12  feet  11  inches,  and  the  circumference  of  its  bass  5  feet  3 
inches  ? 

Ans.,  28.3308  cubic  feet. 

PROBLEM  III. 
(121.)  To  find  the  surface  of  a  cone. 

RULE. 

Multiply  the  circumference  of  the  base  by  half  the  side  for 
the  convex  surface  ;  to  which  add  the  area  of  the  base  ivhen 
(he  entire  surface  is  required 

See  Geometry.  Prop.  3,  B.  X. 


MENSURATION   OF    SOLIDS.  85 

Ex.  1.  What  is  the  entire  surface  of  a  cone  whose  side  is  10 
feet  and  the  diameter  of  its  base  2  feet  3  inches  ? 

Am.,  39.319  sqiure  feet. 

Ex.  2.  What  is  the  entire  surface  of  a  cone  whose  side  ia  15 
feet,  and  the  circumference  of  its  base  8  feet  ? 

Ans.,  65.093  square  feet. 

PROBLEM  IV. 
(122.)   To  find  the  solidity  of  a  cone. 

RULE. 

Multiply  the  area  of  the  base  by  one  third  of  the  altitude. 
See  Geometry,  Prop.  5,  B.  X. 

Ex.  1.  What  is  the  solidity  of  a  cone  whose  altitude  is  12 
feet,  and  the  diameter  of  its  base  2  J  feet  ? 

Ans.,  19.635  cubic  feet. 

Ex.  2.  What  is  the  solidity  of  a  cone  whose  altitude  is  25 
fcet,  and  the  circumference  of  its  base  6  feet  9  inches  ? 

Ans. 

PROBLEM  V. 
<123.)  To  find  the  surface  of  a  frustum  of  a  cont. 

RULE. 

Multiply  half  the  side  by  the  sum  of  the  circumferences  of 
the  two  bases  for  the  convex  surface  ;  to  this  add  the  areas 
of  the  two  bases  when  the  entire  surface  is  required. 
See  Geometry,  Prop.  4,  B.  X. 

Ex.  1.  What  is  the  entire  surface  of  a  frustum  of  a  cone, 
the  diameters  of  whose  basrs  are  9  feet  and  5  feet,  and  whose 
hide  is  16  feet  9  inches  ? 

Ans.)  451.6036  square  feet. 

Ex.  2.  What  is  the  convex  surface  of  a  frustum  of  a  cone 
whose  side  is  10  feet,  and  the  circumferences  of  its  bases  6 
feet  and  4  feet  ? 

Ans.^  50  square  feet. 

PROBLEM  VI. 
1124.)  To  find  the  solidity  of  a  frustum  of  a  cone. 


#6  "TRIGONOMETRY. 

RULE 

Add  together  the  areas  of  the  two  base*,  and  a  mean  prv- 
portional  between  them,  and  multiply  the  sum  by  one  third 
of  the  altitude. 

See  Geometry,  Prop.  6,  B.  X. 

If  we  put  R  and  r  for  the  radii  of  the  two  bases,  then  TrR' 
will  represent  the  area  of  one  base,  Trr*  the  area  of  the  other, 
and  TrRr  the  mean  proportional  between  them.  Hence,,  if  wo 
represent  the  height  of  the  frustum  by  h,  its  solidity  will  be 


Ex.  1.  What  is  the  solidity  of  a  frustum  of  a  cons  whose 
altitude  is  20  feet,  the  diameter  of  the  greater  end  5  feet,  and 
that  of  the  less  end  2  feet  6  inches  ? 

Ans.,  229.074  cubic  feet 

Ex.  2.  The  length  of  a  mast  is  60  feet,  its  diameter  at  the 
greater  end  is  20  inches,  and  at  the  less  end  12  inches  :  what 
is  its  solidity  ?  Ans.,  85.521  cubic  feet 

PROBLEM  VII. 
(125.)  To  find  the  surface  of  a  sphere. 

RULE. 

Multiply  the  diameter  by  the  circumference  of  a  great  cir- 
cle ;  or,  Multiply  the  square  of  the  diameter  by  3.14159 

See  Geometry,  Prop.  7,  B.  X. 

Ex.  1.  Required  the  surface  of  the  earth,  its  diameter  be- 
ing 7912  miles.  Ans.,  196,662,896  square  miles, 

Ex.  2.  Required  the  surface  of  the  moon,  its  circumference 
being  6786  miles.  Ans. 

PROBLEM  VIII. 
(126.)  To  find  trie  solidity  of  a  sphere. 

RULE. 

Multiply  the  surface  by  one  third  of  the  radius  ;  or,  Mul* 
tiply  the  cube  of  the  diameter  by  ITT]  that  is,  by  0.5238. 
See  Geometry,  Prop.  8,  B.  X. 
Where  creal  accuracy  is  required,  the  value  cf  $w  must  ba 


MENSURATION   OF    S o L i D s. 


taken  to  more  than  four  decimal  places.     Its  value,  correct  to 
ten  decimal  places,  is  .52359,87756. 

Ex.  1.  What  is  the  solidity  of.  the  earth,  if  it  be  a  sphere 
7912  miles  in  diameter  ? 

Ans.,  259,332,805,350  cubic  miles. 

Ex.  2.  If  the  diameter  of  the  moon  be  2160  miles,  what  is 
its  solidity?  Ans. 

PROBLEM  IX. 
(127.)  To  find  the  surface  of  a  spherical  zone. 

RULE. 

Multiply  the  altitude  of  the  zone  by  the  circumference  o)  a 
great  circle  of  the  sphere. 

See  Geometry,  Prop.  7,  Cor.  1,  B.  X. 

Ex.  1.  If  the  diameter  of  the  earth  be  7912  miles,  what  is 
the  surface  of  the-  torrid  zone,  extending  23°  27'  36"  on  each 
«ide  of  the  equator  ? 

Ans.,  78,293,218  square  miles 

Let  PEP'Q,  represent  a  meridian  of  the  earth;  EQ,  the 
equator ;  P,  P'  the  poles  ;  AB  one  of  the 
tropics,  and  GH  one  of  the  polar  circles. 
Then  PK  will  represent  the  height  of  one 
of  the  frigid  zones,  KD  the  height  of  one 
of  the  temperate  zones,  and  CD  half  the 
height  of  the  torrid  zone. 

Each  of  the  angles  ACE,  CAD,  and 
GCK  is  equal  to  23°  27'  36". 

Tn  the  right-angled  triangle  ACD, 

R  :  AC  :  :  sin.  CAD  :  CD. 
Also,  in  the    ight-angled  triangle  CGrK, 

R :  CO  : :  cos.  GCK  :  CK, 
Then  PK=PC-KC. 

Where  great  accuracy  is  required,  the  sine  and  cosine  ol 
23°  27'  36"  must  be  taken  to  more  than  six  decimal  places 
The  following  values  are  correct  to  ten  decimal  places : 
Natural  sine  of  23°  27'  36"=. 39810,87431. 
"    cosine  of  23°  27'  36" :=. 91733,82302. 


88  TRIGONOMETRY. 

Ex.  2.  If  the  polar  circle  extends  23°  27'  36"  from  the  pole 
find  the  convex  surface  of  either  frigid  zone. 

Ans.,  8,128,252  square  miles. 

Ex.  3.  On  the  same  suppositions,  find  the  surface  of  each  oi 
the  temperate  zones. 

Ans..  51,056,587  square  miles. 

PROBLEM  X. 

(128.)  To  find  the  solidity  of  a  spherical  segment  with  one 
base, 

RULE. 

Multiply  half  the  height  of  the  segment  by  the  area  of  the 
base,  and  the  cube  of  the  height  by  .5236,  and  add  the  two 
products. 

See  Geometry,  Prop.  9,  B.  X. 

Ex.  1.  What  is  the  solidity  of  either  frigid  zone,  supposing 
the  earth  to  be  7912  miles  in  diameter,  the  polar  circles  ex- 
tending 23°  27'  36"  from  the  poles  ? 

Ans.,  1,292,390,176  cubic  miles. 

(129.)  The  solidity  of  a  spherical  segment  of  two  bases  is 
(he  difference  between  two  spherical  segments,  each  having  a 
single  base. 

Ex.  2.  On  the  same  supposition  as  in  Ex.  1,  find  the  solid- 
ity of  either  temperate  zone. 

Ans.,  55,032,766,543  cubic  miles 
Ex.  3.  Find  the  solidity  of  the  torrid  zone. 

Ans.,  146,682,491,911  cubic  miles 

PROBLEM  XL 
(130.)  To  find  the  area  of  a  spherical  triangle. 

RULE. 

Compute  the  surface  of  the  quadrantal  triangle,  or  one 
eighth  of  the  surface  of  the  sphere.  From  the  sum  of  the 
three  angles  subtract  two  right  angles  ;  divide  the  remainder 
by  90,  and  multiply  the  quotient  by  the  quadrantal  triangh 

Sco  Geometry,  Prop.  20,  B.  IX. 


MENSURATION   OF    SOLIDS.  89 

Ex,  1.  "What  is  the  area  of  a  triangle  on  a  sphere  whose  di 
ameter  is  10  feet,  if  the  angles  are  55°,  60°,  and  85°  ? 

Ans.j  8.7266  square  feet. 

Ex.  2.  If  the  angles  of  a  spherical  triangle  measured  on  the 
surface  of  the  earth  are  78°  4'  10",  59°  50'  54",  and  42°  5'  37", 
what  is  the  area  of  the  triangle,  supposing  the  earth  a  sphere, 
of  which  the  diameter  is  7912  miles  ? 

Ans.j  3110.794  square  miles. 

If  the  excess  of  the  angles  above  two  right  angles  is  ex- 
pressed  in  seconds,  we  must  divide  it  by  90  degrees  also  ex- 
pressed in  seconds  ;  that  is,  by  324,000. 

PROBLEM  XII. 
(131.)  To  find  the  area  of  a  spherical  polygon. 

RULE. 

Compute  the  surface  of  the  quadrantal  triangle.  From 
the  sum  of  all  the  angles  subtract  the  product  of  two  right 
angles  by  the  number  of  sides  less  two  ;  divide  the  remainder 
by  90,  and  multiply  the  quotient  by  the  quadrantal  triangle 

See  G-eometry,  Prop.  21,  B.  IX. 

Ex.  1.  What  is  the  area  of  a  spherical  polygon  of  5  sides  on 
a  sphere  whose  diameter  is  10  feet,  supposing  the  sum  of  tbe 
angles  to  be  640  degrees  ? 

Ans.j  43.633  square  feet 

62°  33'  13" ; 


Q/   ^)fiff  • 

Ex.  2.  The  angles  of  a  spherical  r~~  ^  p,  Q,,  ' 
polygon,  measured  on  the  surface  •<  ^  ^  0  .  _;  ^/;  | 
of  the  earth,  are  ^.0  .Q,  _/;  | 

155°  19'  12". 
Required  the  area  of  the  polygon. 

Ans.t  5690.477  square  miles 


BOOK  IV. 

SURVEYING. 

(182.)  THE  term  Surveying  includes  the  measurement  ol 
heights  and  distances,  the  determination  of  the  area  of  portions 
of  the  earth's  surface,  and  their  delineation  upon  paper. 

Since  the  earth  is  spherical,  its  surface  is  not  a  piano  sur* 
lace,  and  if  large  portions  of  the  earth  are  to  be  measured,  the 
curvature  must  be  taken  into  account ;  but  in  ordinary  sur- 
veying, the  portions  of  the  earth  are  supposed  to  be  so  small 
that  the  curvature  may  be  neglected.  The  parts  surveyed  are 
therefore  regarded  as  plane  figures. 

(133.)  If  a  plummet  be  freely  suspended  by  a  line,  and  al- 
lowed to  come  to  a  state  of  rest,  this  line  is  called  a  vertical 
tine. 

Every  plane  passing  through  a  vertical  line  is  a  vertica. 
plane. 

A  line  perpendicular  to  a  vertical  line  is  a  horizontal  line. 

A  plane  perpendicular  to  a  vertical  line  is  a  horizontal 
plane. 

A  vertical  angle  is  one  the  plane  of  whose  sides  is  vertical 

A  horizontal  angle  is  one  the  plane  of  whose  sides  is  hori- 
zontal. 

An  angle  of  elevation  is  a  vertical  angle  having  one  side 
horizontal  and  the  other  an  amending, 
line,  as  the  angle  BAD. 

An  angle  of  depression  is  a  vertical 
angle  having  one  side  horizontal  and 
the  other  a  descending  line,  as  the  an- 
gle CDA. 

(134.)  When  distances  are  to  be  found 
by  trigonometrical  computation,  it  is  necessary  to  measure  at 
least  one  line  upon  the  ground,  and  also  as  many  angles  as 
may  be  necessary  to  render  three  parts  of  every  triangle 
known 


SURVEYING.  91 

la  the  measurement  of  lines,  the  unit  communly  employed 
ty  surveyors  is  a  chain  four  rods  or  sixty-six  feet  in  length, 
called  Gunter's  Chain,  from  the  name  of  the  inventor.  This 
chain  is  divided  into  100  links.  Sometimes  a  half  chain  i* 
used,  containing  50  links. 

Hence,         1  chain—  100  links    =66  feet; 
Irod     =    25  links    =16J  feet; 
1  link   =7.92  inches  =     f  of  a  foot  nearly. 

(135.)  To  measure  a  horizontal  line. 

To  mark  the  termination  of  the  chain  in  measuring,  ten  iro* 
pins  should  be  provided,  about  a  foot  in  length. 

Let  the  person  who  is  to  go  foremost  in  carrying  the  chain, 
and  who  is  called  the  leader,  take  one  end  of  the  chain  and  the 
ten  pins ;  and  let  another  person  take  the  other  end  of  the 
chain,  and  hold  it  at  the  beginning  of  the  line  to  be  measured. 
When  the  leader  has  advanced  until  the  chain  is  stretched 
tight,  he  must  set  down  one  pin  at  the  end  of  the  chain,  the 
other  person  taking  care  that  the  chain  is  in  the  direction  of 
the  line  to  be  measured.  Then  measure  a  second  chain  in  the 
same  manner,  and  so  on  until  all  the  marking  pins  are  ex- 
hausted.  A  record  should  then  be  made  that  ten  chains  ha\re 
been  measured,  after  which  the  marking  pins  should  be  re- 
turned to  the  leader,  and  the  measurement  continued  as  be- 
fore until  the  whole  line  has  been  passed  over. 

It  is  generally  agreed  to  refer  all  surfaces  to  a  horizontal 
plane.  Hence,  when  an  inclined  surface,  like  the  side  of  a 
hill,  is  to  be  measured,  the  chain  should  be  maintained  in  a 
horizontal  position.  For  this  purpose,  in  ascending  a  hill,  the 
hind  end  of  the  chain  should  be  raised  from  the  ground  until 
it  is  on  a  level  with  the  fore  end,  and  should  be  held  vertically 
over  the  termination  of  the  preceding  chain.  In  descending  a 
hill,  the  fore  end  of  the  chain  should  be  raised  in  the  same 
manner. 


INSTRUMENTS  FOR  MEASURING  ANGLES. 

In  measuring  angles,  some  instrument  is  used  which  con- 
tains a  portion  of  a  graduated  circle  divided  into  degrees  and 
minutes  These  instruments  may  be  adapted  to  measuring 


>2  TRIGONOMETRY. 

either  horizontal  o:  vertical  angles.     The  instrument  most  fro 
quently  employed  for  measuring  horizontal  angles  is  called 

THE  SURVEYOR'S  COMPASS. 

(136.)  The  piincipal  parts  of  this  instrument  are  a  compas&. 
box,  a  magnetic  needle,  two  sights,  and  a  stand  for  its  support, 
The  compass-box,  ABC,  is  circular,  generally  about  six  inches 
in  diameter,  and  at  its  center  is  a  small  pin  on  which  the  mag- 
netic needle  is  balanced.  The  circumference  of  the  box  is  di- 
vided into  degrees,  and  sometimes  to  half  degrees  ;  and  the  de- 
grees are  numbered  from  the  extremities  of  a  diameter  both 
ways  to  90°.  The  sights,  DE,  FGr,  are  placed  at  right  angles 


to  the  plane  of  the  graduated  circle,  and  in  each  of  these  there 
is  a  large  and  small  aperture  for  convenience  of  observation 
The  instrument,  when  used,  is  mounted  on  a  tripod,  or  a  single 
staff  pointed  with  iron  at  the  bottom,  so  that  it  may  be  firmly 
placed  in  the  ground. 

Sometimes  two  spirit  levels,  H  and  K,  are  attached,  to  indi- 
cate when  the  plane  of  the  graduated  circle  is  brought  into  a 
horizontal  position. 

(137.)  When  the  magnetic  needle  is  supported  so  as  to  turn 
freely,  and  is  allowed  to  come  to  a  state  of  rest,  the  direction 
it  assumes  is  called  the  magnetic  meridian,  one  end  of  the 
needle  indicating  the  north  point  and  the  other  the  south. 

A  horizontal  line  perpendicular  to  a  meridian  is  an  east  an;i 
west  line 


SURVEYING. 


All  the  meridians  passing  through  a  survey  of  moderate  ex- 
tent, are  considered  as  straight  lines  parallel  to  each  other. 

The  bearing  or  course  of  a  line  is  the  angle  which  it  makes 
with  a  meridian  passing  through  one  end ;  and  it  is  reckoned 
from  the  north  or  south  point  of  the  horizon,  toward  the  east 
or  west. 

Thus,  if  NS  represent  a  meridian,  and  the  angle  NAB  is  40°, 
then  the  bearing  of  AB  from  the  point  A  is  XT 

40°  to  the  west  of  north,  and  is  written  N.  40° 
\V.,  and  read  north  forty  degrees  west.  ] 


The  reverse  bearing  of  a  line  is  the  bearing 
taken  from  the  other  end  of  the  line. 

The  forward  bearing  and  reverse  bearing 
of  a  line  are  equal  angle?,  but  lie  between  di- 
rectly opposite  points.  Thus,  if  the  bearing 
of  AB  from  A  is  N.  40°  W.,  the  bearing  of  the 
same  line  from  B  is  S.  40°  E. 

(138.)  For  measuring  vertical  angles,  the  instrument  com- 
monly  used  is 

A  QUADRANT. 

It  consists  of  a  quarter  of  a  circle,  usually  made  of  brass, 
and  its  limb,  AB,  is  divided  into 
degrees  and  minutes,  numbered 
from  A  up  to  90°.  It  is  furnish- 
ed either  with  a  pair  of  plain 
sights  or  with  a  telescope,  CD, 
which  is  to  be  directed  toward 
the  object  observed.  A  plumb 
line,  CE,  is  suspended  from  the 
center  of  the  quadrant,  and  in- 
dicates when  the  radius  CB  is 
brought  into  a  vertical  position. 

To  measure  the  angle  of  elevation,  for  example,  of  the  top 
of  a  tower,  point  the  telescope,  CD,  toward  the  tower,  keeping 
the  radius,  CB,  in  a  vertical  position  by  means  of  the  plumb 
line,  CE.  Move  the  telescope  until  the  given  object  is  seen  in 
the  middle  of  the  field  of  view.  The  center  of  the  field  is  in- 
ilicated  by  two  wires  placed  in  the  focus  of  the  object-glass  of 


TRIGONOMETRY 


the  telescope,  one  wire  being  vertical  and  the  other  horizonta 
When  the  horizontal  wire  is  made  to  coincide  with  the  sum 
mit  of  the  tower,  the  angle  of  elevation  is  shown  upon  the  arc 
AB  by  means  of  an  index  which  moves  with  the  telescope. 

As  the  arc  is  not  commonly  divided  into  parts  smaller  than 
half  degrees,  when  great  accuracy  is  required,  some  contriv* 
ance  is  needed  for  obtaining  smaller  fractions  of  a  degree. 
This  is  usually  effected  by  a  vernier. 

(139.)  A  Vernier  is  a  scale  of  small  extent,  graduated  iu 
such  a  manner  that,  being  moved  by  the  side  of  a  fixed  scale, 
we  are  enabled  to  measure  minute  portions  of  this  scale.  The 
length  of  this  movable  scale  is  equal  to  a  certain  number  of 
parts  of  that  to  be  subdivided,  but  it  is  divided  into  parts  one 
more  or  one  less  than  those  of  the  primary  scale  taken  for  the 
length  of  the  vernier.  Thus,  if  we  wish  to  measure  hundredths 
of  an  inch,  as  in  the  case  of  a  barometer,  we  first  divide  an 
inch  into  ten  equal  parts.  "We  then  construct  a  vernier  equal 
in  length  to  11  of  these  divisions,  but  divide  it  into  10  equal 
jarts,  by  which  means  each  division  on  the  vernier  is  TVth 
longer  than  a  division  of  the  primary  scale. 

Thus,  let  AB  be  the  upper  end  of  a  barometer  tube,  the  mer 
sury  standing  at  the  point  C  ;  the  scale  is 
rlivided  into  inches  and  tenths  of  an  inch, 
and  the  middle  piece,  numbered  from  1 
to  9,  is  the  vernier  that  slides  up  and 
down,  having  10  of  its  divisions  equal  to 
11  divisions  of  the  scale,  that  is,  to  }  jths 
of  an  inch.  Therefore,  each  division  of 
the  vernier  is  Ty  „  ths  of  an  inch  ;  or  one 

division  of  the  vernier  exceeds  one  divi-  !  IIIHIIil  6HH  29 
sion  of  the  scale  by  Tj¥th  of  an  inch. 
Now,  as  the  sixth  division  of  the  vernier 
(in  the  figure)  coincides  with  a  division 
of  the  scale,  the  fifth  division  of  the  ver- 
nier will  stand  th  of  an  inch  above  the  nearest  division  o/ 


the  scale  ;  the  fourth  division  fifths  of  an  inch,  and  the  to* 
of  the  vernier  will  be  T£wths  of  an  inch  above  the  next  lowei 
division  of  the  scale  ;  i.  e.,  the  top  of  the  vernier  coincides  with 
29  G6  inches  upon  the  scalo.  in  practice,  therefore,  we  ob- 


SURVEYING.  93 

serve  what  division  of  the  vernier  coincides  with  a  division  of 
the  scale  ;  this  will  show  the  hundredths  of  an  inch  to  be  added 
to  the  tenths  next  below  the  vernier  at  the  top. 

A  similar  contrivance  is  applied  to  graduated  circles,  to  ob- 
tain the  value  of  an  arc  with  greater  accuracy.  If  a  circle  ia 
graduated  to  half  degrees,  or  30',  and  we  wish  to  measure  sin- 
gle minutes  by  the  vernier,  we  take  an  arc  equal  to  31  divi- 
sions upon  the  limb,  and  divide  it  into  30  equal  parts.  Then 
each  division  of  the  vernier  will  be  equal  to  £  Jths  of  a  degree, 
while  each  division  of  the  scale  is  f  £ths  of  a  degree.  That  is, 
each  space  on  the  vernier  exceeds  one  on  the  limb  by  1'. 

In  order,  therefore,  to  read  an  angle  for  any  position  of  the 
vernier,  we  pass  along  the  vernier  until  a  line  is  found  coin- 
ciding with  a  line  of  the  limb.  The  number  of  this  line  from 
the  zero  point  indicates  the  minutes  which  are  to  be  added  to 
the  degrees  and  half  degrees  taken  from  the  graduated  circle. 
Sometimes  a  vernier  is  attached  to  the  common  surveyor's 
compass. 

(140.)  An  instrument  in  common  use  for  measuring  both 
horizontal  and  vertical  angles  is 

THE  THEODOLITE. 

The  theodolite  has  two  circular  brass  plates,  C  and  D  (see  fig. 
next  page),  the  former  of  which  is  called  the  vernier  plate,  and 
the  latter  the  graduated  limb.  Both  have  a  horizontal  motion 
about  the  vertical  axis,  E.  This  axis  consists  of  two  parts,  ono 
external,  and  the  other  internal ;  the  former  secured  to  the 
graduated  limb,  D,  and  the  latter  to  the  vernier  plate,  C,  so  that 
the  vernier  plate  turns  freely  upon  the  lower.  The  edge  of  the 
lower  plate  is  divided  into  degrees  and  half  degrees,  and  this 
is  subdivided  by  a  vernier  attached  to  the  upper  plate  into 
single  minutes.  The  degrees  are  numbered  from  0  to  360. 

The  parallel  plates,  A  and  B,  are  held  together  by  a  ball 
which  rests  in  a  socket.  Four  screws,  three  of  which,  #,  a,  a, 
are  shown  in  the  figure,  turn  in  sockets  fixed  to  the  lower  plate, 
while  their  heads  press  against  the  under  side  of  the  uppc* 
plate,  by  which  means  the  instrument  is  leveled  for  observe 
tnn.  The  whole  rests  upon  a  tripod,  which  is  firmly  aU&clit>d 
to  the  body  of  the  instrument. 


TRIGONOMETRY. 


To  the  vernier  plate,  two  spirit-levels,  c,  c,  are  attached  ai 
right  angles  to  each  other,  to  determine  when  the  graduated 
limb  is  horizontal.  A  compass,  also,  is  placed  at  F.  Two 
frames,  one  of  which  is  seen  at  N,  support  the  pivots  of  tho 
horizontal  axis  of  the  vertical  semicircle  KL,  on  which  the  tel- 
escope, GrH,  is  placed.  One  side  of  the  vertical  arc  is  divided 
into  degrees  and  half  degrees,  and  it  is  divided  into  single  min- 
utes "by  the  aid  of  its  vernier.  The  graduation  commences  at 
the  middle  of  the  arc,  and  reads  "both  ways  to  90°.  Under  and 
parallel  to  the  telescope  is  a  spirit-level,  M,  to  show  when  tho 
telescope  is  brought  to  a  horizontal  position.  To  enable  us  to 
direct  the  telescope  upon  an  object  with  precision,  two  lines 
called  wires  are  fixed  at  right  angles  to  each  other  in  the  focus 
of  the  telescope. 

To  measure  a  Horizontal  Angle  with  the  Theodolite. 
(141.)  Place  the  instrument  exactly  over  the  station  from 
which  the  angle  is  to  be  measured ;  then  level  the  instrument 
by  means  of  the  screws,  a,  #,  bringing  the  telescope  over  each 
pair  alternately  until  the  two  spirit-levels  on  the  vernier  plate 
retain  their  position,  while  the  instrument  is  turned  entirely 
round  upon  its  ay  is.  Direct  the  telescope  to  one  cf  the  object- 


SURVEYING.  97 

to  be  observed,  moving  it  until  the  cross-wires  and  abject  co- 
incide. Now  read  off  the  degrees  upon  the  graduated  limb, 
and  the  minutes  indicated  ""ry  the  vernier.  Next,  release  the 
upper  plate  (leaving  the  graduated  limb  undisturbed),  and 
move  it  round  until  the  telescope  is  directed  to  the  second  ob- 
ject; and  make  the  cross-wires  bisect  this  object,  as  was  done 
by  the  first.  Again,  read  off  the  vernier  ;  the  difference  bo- 
tween  this  and  the  former  reading  will  be  the  angle  required. 
The  magnetic  bearing  of  an  object  is  determined  by  simply 
reading  the  angle  pointed  out  by  the  compass-needle  when  the 
object  is  bisected. 

To  measure  an  Angle  of  Elevation  with  the  Theodolite. 

(142.)  Direct  the  telescope  toward  the  given  object  so  that 
it  may  be  bisected  by  the  horizontal  wire,  and  then  read  off 
the  arc  upon  the  vertical  semicircle.  After  observing  the  ob- 
ject with  the  telescope  in  its  natural  position,  it  is  well  to  re* 
volve  the  telescope  in  its  supports  until  the  level  comes  upper- 
most, and  repeat  the  observation.  The  mean  of  the  two  meas . 
ures  may  be  taken  as  the  angle  of  elevation. 

By  the  aid  of  the  instruments  now  described,  we  may  de- 
termine the  distance  of  an  inaccessible  object,  and  its  height 
above  the  surface  of  the  earth. 

HEIGHTS  AND  DISTANCES. 
PROBLEM  I. 

(143.)  To  determine  the  height  of  a  vertical  object  situated 
i:n  a  horizontal  plane. 

Measure  from  the  object  to  any  convenient  distance  in  a 
slraight  line,  and  then  take  the  angle  of  elevation  subtended 
hy  the  object.  C, 

If  we  measure  the  distance  DE ,  and 
the  angle  of  elevation  CDE,  there  will 
be  given,  in  the  right-angled  triangle 
CDE,  the  base  and  the  angles,  to  find 
the  perpendicular  CE  (Art.  46).  To 
this  we  must  add  the  height  of  the  in- 
strument, to  obtain  the  entire  height 
i>f  the  object  aT;*ove  the  plane  AB. 

Gr 


T  R  I  G  O  IV  O  M  E  T  R  v 

Ex.  1.   Having  measured  AB  equal    to  100  feet  from  the 


c 


bottom  of  a  tower  on  a  horizontal 
plane,  I  found  the  angle  of  elevation, 
CDE,  of  the  cop  to  be  47°  30',  the 
center  of  the  quadrant  being  five 
feet  above  the  ground.  What  is  the  / 

height  of  the  tower  1  / 

B,  :  tang.  GDE  :  :  DE  :  CE  =  109.13. 
To  which  add  five  feet,  and  we  obtain      f 


-U- 


the  height  of  the  tower,  114.13  feet.     A 

Ex.  2.  From  the  edge  of  a  ditch  18  feet  wide,  surrounding 
a  fort,  the  angle  of  elevation  of  the  wall  was  found  to  be  62° 
40'.  Required  the  height  of  the  wall,  and  the  length  of  a  lad- 
der necessary  to  reach  from  my  station  to  the  top  of  it. 

Ans.  The  height  is  34.82  feet.     Length  of  ladder,  39.20  feet. 

PROBLEM  II. 

(144.)  To  find  the  distance  of  a  vertical  object  whor-e  hetgitt 
is  known. 

Measure  the  angle  of  elevation,  and  we  shall  have  given  the 
angles  and  perpendicular  of  a  right-angled  triangle  to  find  the 
base  (Art.  46). 

Ex.  1.  The  angle  of  elevation  of  the  top  of  a  tower  whoso 
height  was  known  to  be  143  feet,  was 
found  to  be  35°.     What  was  its  dis- 
tance 1 

Here  we  have  given  the  angles  of  the 
triangle  ABC,  and  the  side  CB,  to  find 
AB. 

Ans.,  204.22  feet. 

i  If  the  observer  were  stationed  at  the 
top  of  the  tower  BC,  he  might  find  the  length  of  the  base  AB 
by  measuring  the  angle  of  depression  DC  A,  which  is  equal  to 

BAG: 

Ex.  2.  From  the  top  of  a  ship's  mast,  which  was  80  feet 
above  the  water,  the  angle  of  depression  of  another  ship's  hull 
*«-as  found  to  be  20°.  What  was  its  distance  1 

Ans.,  219.80  feet 


SURVEYING. 


99 


PROBLEM  III. 

(145.)  To  find  the  height  of  a  vertical  object  standing  on 
an  inclined  plane. 

Measure  the  distance  from  the  object  to  any  convenient  sta- 
tion, and  observe  the  angles  which  the  base-line  makes  with 
lines  drawn  from  its  two  ends  to  the  top  of  the  object. 

If  we  measure  the  base-line  AB,  and  the  two  angles  ABC, 
BAG,  then,  in  the  triangle  ABC, 
we  shall  have  given  one  side  and 
the  angles  to  find  BC. 

Ex.  1.  Wanting  to  know  the 
height  of  a  tower  standing  on  an 
inclined  plane,  BD,  I  measured 
from  the  bottom  of  the  tower  a 
distance,  AB,  equal  to  165  feet ; 
also  the  angle  ABC,  equal  to 
107°  18',  and  the  angle  BAG, 
equal  to  33°  35'.  Required  the 
height  of  the  object. 

sin.  ACB  :  AB  :  :  sin.  BAG  :  BC=144.66  feet. 

The  height,  BC,  may  also  be  found  by  measuring  the  dis- 
tances BA,  AD,  and  taking  the  angles  BAG,  BDC.  The  dif- 
ference between  the  angles  BAG  and  BDC  will  be  the  angle 
ACD.  There  will  then  be  given,  in  the  triangle  DAG,  one 
side  and  all  the  angles  to  find  AC  ;  after  which  we  shall  have, 
in  the  triangle  ABC,  two  sides  and  the  included  angle  to  find 
BC. 

Ex.  2.  A  tower  standing  on  the  top  of  a  declivity,  I  meas- 
ured 75  feet  from  its  base,  and  then  took  the  angle  BAG,  47r 
60' ;  going  on  in  the  same  direction  40  feet  further,  I  took  the 
angle  BDC,  38°  30'.  What  was  the  height  of  the  tower  ? 

Am.,  117.21  feet. 

PROBLEM  IY. 

(146.)  To  find  the  distance  of  an  inaccessible  object. 

Measure  a  horizontal  base-line,  and  also  the  angles  between 
tills  line  and  lines  drawn  from  each  station  to  the  object.  Lei 
0  -<3  the  obieot  inaccessible  from  A  and  R  Then>  if  the  din 


100 


T  ii  >  c  o  N  o  M  L  r  R  y. 


sin.  C  :  AB 


tance  between  the  stations  A  and  B  be  measured,  as  also  the 
angles  at  A  and  B,  there  will  be  given,  in 
the  triangle  ABC,  the  side  AB  and  the  an- 
gles, to  find  AC  and  BC,  the  distances  of  the 
object  from  the  two  stations. 

Ex.  1.  Being  on  the  side  of  a  river,  and 
wanting  to  know  the  distance  to  a  house 
which  stood  on  the  other  side,  I  measured  A  B 

400  yards  in  a  right  line  by  the  side  of  the  river,  and  founr 
that  the  two  angles  at  the  ends  of  this  line,  formed  by  the 
other  end  and  the  house,  were  73°  15'  and  68°  2'.  What  was 
the  distance  between  each  station  and  the  house  ? 

The  angle  C  is  found  to  be  38°  43'.     Then 

sin.  A  :  BC-612.38; 
sin.  B  :  AC-593.09. 

Ex.  2.  Two  ships  of  war,  wishing  to  ascertain  their  distance 
from  a  fort,  sail  from  each  other  a  distance  of  half  a  mile,  when 
they  find  that  the  angles  formed  between  a  line  from  one  tc 
the  other,  and  from  each  to  the  fort,  are  85°  15'  and  63°  4S' 
What  are  the  respective  distances  from  the  fort  ? 

Ans.,  4584.52  and  4596.10  yards. 

PROBLEM  V. 

(147.)  To  find  the  distance  between  two  objects  separated 
by  an  impassable  barrier. 

Measure  the  distance  from  any  convenient  station  to  each 
of  the  objects,  and  the  angle  included  between  those  lines. 

If  we  wish  to  know  the  distance  between  the  places  C  and 
B,  both  of  which  are  accessible,  but  sep- 
arated from  each  other  by  water,  we  may   c 
measure  the  lines  AC  and  AB,  and  also 
the  angle  A.     We  shall  then  have  given 
two  sides  of  a  triangle  and  the  included 
angle  to  find  the  third  side. 

Ex.  1.  The  passage  between  the  two 
nbjects  C  and  B  being  obstructed,  I  measured  from  A  to  C  735 
rods,  and  from  A  to  B  840  rods ;  also,  the  angle  A,  equal  tn 
55°  40'.  What  is  the  distance  of  the  places  C  and  B  ? 

Ans.,  741.21  rod- 


SURVEYING.  101 

Ex.  2.  In  order  to  find  the  distance  between  two  cbpcts,  C 

-id  B,  which  could  not  be   directly  measured,  I  measured 

i  om  C  to  A  652  yards,  and  from  B  to  A  756  yards ;  also,  the 

angle  A  equal  to  142°  25'.     What  is  the  distance  between  tho 

objects  C  and  B  ? 

Ans. 

PROBLEM  VI. 

(148.)  To  find  the  height  of  an  inaccessible  object  above  a 
horizontal  plane. 

First  Method. — Take  two  stations  in  a  vertical  plane  pass- 
ing  through  the  top  of  the  object ;  measure  the  distance  be- 
tween the  stations  and  the  angle  of  elevation  at  each. 

If  we  measure  the  base  AB,  and  the  angles  DAG,  DBC, 
then,  since  CBA  is  the  sup- 
plement  of  DBC,  we  shall 
have,  in  the  triangle  ABO, 
one  side  and  all  the  angles 

to  find  BC.     Then,  in  the  3b 

right-angled  triangle  DBC,  we  shall  have  the  hypothenuse  and 
the  angles  to  find  DC.      . 

Ex.  1.  What  is  the  perpendicular  height  of  a  hill  whose  an- 
gle of  elevation,  taken  at  the  bottom  of  it,  was  46° ;  and  100 
yards  farther  off,  on  a  level  with  the  bottom  of  it,  the  angle 
was  31°  ? 

Ans.,  143.14  yards. 

Ex.  2.  The  angle  of  elevation  of  a  spire  I  found  to  be  58°, 
*nd  going  100  yards  directly  from  it,  found  the  angle  to  be 
only  32°.  What  is  the  height  of  the  spire,  supposing  the  in- 
strument to  have  been  five  feet  above  the  ground  at  each  ob- 
Rervation  ? 

Ans.,  104.18  yards. 

(149.)  Second  Method. — Measure  any  convenient  base-line, 
also  the  angles  between  this  base  and  lines  drawn  from  each 
of  its  extremities  to  the  foot  of  the  object,  and  the  angle  of 
elevation  at  one  of  the  stations. 

Let  DC  be  the  given  object.  If  we  measure  the  horizontal 
base-line  AB,  and  the  angles  CAB,  CBA,  we  can  compute  tha 
iistanco  BC.  Also,  if  we  observe  the  angle  of  elevation  CBD, 


IOL 


T  R  1  G  O  N  D  M  E  T  R  Y. 


we  shall  ha/e  given,  in  the  right-angled  triangle   BCD,  tka 
base  and  angles  to  find  the  perpen- 
dicular. 

Ex.  1.  Being  on  one  side  of  a  river, 
and  wanting  to  know  the  height  of  a  * 
spire  on  the  other  side,  I  measured 
500  yards,  AB,  along  the  side  of  the 
river,  and  found  the  angle  ABC =74° 
14',  and  BAC-490  23' ;  also,  the  an- 
gle of  elevation  CBD=11°  15'.  Required  the  height  of  th« 


Ans.,  271.97  feet. 
Ex.  2.  To  find  the  height  of  an  inaccessible  castle,  I  meas- 
ured a  line  of  73  yards,  and  at  each  end  of  it  took  the  angle  of 
position  of  the  object  and  the  other  end,  and  found  the  one  to  be 
90°,  and  the  other  61°  45' ;  also,  the  elevation  of  the  castle  from 
the  latter  station,  10°  35'.  Required  the  height  of  the  castle 

Ans.,  86.45  feet. 

PROBLEM  VII. 

(150.)  To  find  the  distance  between  two  inaccessible  objects 

Measure  any  convenient  base-line,  and  the  angles  between 
this  base  and  lines  drawn  from  each  of  its  extremities  to  each 
of  the  objects. 

Let  C  and  D  be  the  two  inaccessible  objects.  If  we  meas- 
nre  a  base-line,  AB,  and  the  an- 
gles DAB,  DBA,  CAB,  CBA, 
then,  in  the  triangle  DAB,  we 
shall  have  given  the  side  AB 
and  all  the  angles  to  find  BD ; 
also,  in  the  triangle  ABC,  we 
shall  have  one  side  and  all  the 
angles  to  find  BC ;  and  then,  in 
the  triangle  BCD,  we  shall  have  two  sides,  BD,  BC,  with  thfl 
included  angle,  to  find  DC. 

Ex.  1.  "Wanting  to  know  the  distance  between  a  house  and 
a  mill,  which  were  separated  from  me  by  a  river,  I  measured 
a  base-line,  AB,  300  yards,  and  found  the  angle  CAB=58° 
20',  CAD-370,  ABD=53°  30',  DEC  =  45°  15'.  What  is  tta 
distance  of  the  house  from  the  mill  ?  Ans.,  479.80  yards. 


SURVEYING  103 


Ex.  2.  Wanting  to  know  the  distance  between  two  inaccessi 
lie  objects,  C  and  D,  I  measured  a  base-line,  AB,  28.76  rods, 
and  found  the  angle  CAB -33°,  CAD =66°,  DBA=59°  45', 
and  BBC =76°.  What  is  the  distance  from  C  to  D  ? 

Ans.,  97.C96  rods 

THE  DETERMINATION  OF  AREAS. 

(151.)  The  area  or  content  of  a  tract  of  land  is  the  horizon- 
tal surface  included  within  its  boundaries. 

When  the  surface  of  the  ground  is  broken  and  uneven,  it  is 
very  difficult  to  ascertain  exactly  its  actual  surface.  Hence 
it  has  been  agreed  to  refer  every  surface  to  a  horizontal  plane  ; 
and  for  this  reason,  in  measuring  the  boundary  lines,  it  is  nec- 
essary to  reduce  them  all  to  horizontal  lines. 

The  measuring  unit  of  surfaces  chiefly  employed  by  survey- 
ors is  the  acre,  or  ten  square  chains. 

One  quarter  of  an  acre  is  called  a  rood. 

Since  a  chain  is  four  rods  in  length,  a  square  chain  contains 
sixteen  square  rods  ;  and  an  acre,  or  ten  square  chains,  con- 
tains 160  square  rods.  Square  rods  are  called  perches.  The 
area  of  a  field  is  usually  expressed  in  acres,  roods,  and  perches, 
designated  by  the  letters  A.,  R.,  P. 

When  the  lengths  of  the  bounding  lines  of  a  field  are  given 
in  chains  and  links,  the  area  is  obtained  in  square  chains  and 
square  links.  Now,  since  a  link  is  T|7  of  a  chain,  a  square 
link  will  be  T^¥><T£O-  °f  a  square  chain;  that  is,  To-}o<r  °f  a 
chain.  Hence  we  have  the  following 

*• 

TABLE. 

1  square  chain= 10,000  square  links. 

1  acre=10  square  chains =100, 000  square  links. 

1  acre =4  roods =160  perches. 

If,  then,  the  linear  dimensions  are  links,  the  area  will  be  ex- 
pressed in  square  links,  and  may  be  reduced  to  square  chains 
by  cutting  off  four  places  of  decimals  ;  if  Jive  places  be  cut  off, 
the  remaining  figures  will  be  acres.  If  the  decimal  part  of  an 
Bcre  be  multiplied  by  4,  it  will  give  the  roods,  and  the  result- 
ing decimal,  multiplied  by  40,  will  give  the  perches. 


i04  TRIGONOMETRY. 

(152,)  The  difference  of  latitude,  or  the  northing  or 
ing-  of  a  line,  is  the  distance  that  one  end  is  further  north  ot 
south  than  the  other  end. 

Thus,  if  NS  be  a  meridian  passing  through  the  end  A  of  thu 
line  AB,  and  BC  be  perpendicular  to  NS,  then  is  ™ 

AC  the  difference  of  latitude,  or  northing  of  AB. 

The  departure,  or  the  easting'  or  westing  of  a 
Jne,  is  the  distance  that  one  end  is  further  east  or 
west  than  the  other  end. 

Thus  BC  is  the  departure  or  westing  of  the  line 
AB. 

It  is  evident  that  the  distance,  difference  of  lat-  _ 

.tude,  and  departure  form  a  right-angled  triangle, 
of  which  the  distance  is  the  hypothenuse. 

The  meridian  distance  of  a  point  is  the  perpendicular  let  falj 
from  the  given  point  on  some  assumed  meridian,  and  is  east  or 
west  according  as  this  point  lies  on  the  east  or  west  side  of  th^ 
meridian. 

The  meridian  distance  of  a  line  is  the  distance  of  the  middle 
point  of  that  line  from  some  assumed  meridian. 

(153.)  When  a  piece  of  ground  is  to  be  surveyed,  we  begin 
at  one  corner  of  the  field,  and  go  entirely  around  the  field, 
measuring  the  length  of  each  of  the  sides  with  a  chain,  and 
their  bearings  with  a  compass. 

Plotting  a  Survey. 

When  a  field  has  been  surveyed,  it  is  easy  to  draw  a  plan 
of  it  on  paper.  For  this  purpose,  draw  a  line  to  represent  the 
meridian  passing  through  the  first  station  ;  then  lay  off  an  an- 
gle equal  to  the  angle  which  the  first  side  of  the  field  makes 
with  the  meridian,  and  take  the  length  of  the  side  from  a  scale 
of  equal  parts.  Through  the  extremity  of  this  side  draw  a 
second  meridian  parallel  to  the  first,  and  proceed  in  the  same 
manner  with  the  remaining  sides.  This  method  will  be  easily 
understood  from  an  example. 

EXAMPLE  1. 

Draw  a  plan  of  a  field  from  the  following  courses  and  <lis> 
tar-cos,  as  given  in  the  field-book 


SURVEYING. 


H)c 


Statfons      |                Bearings. 

Distances. 

1 

N.  45°  E. 

9.30  chains. 

2 

a  60°  E. 

11.85       " 

3 

S.  20°  W. 

5.30       " 

4 

S.  70°  W. 

10.90       « 

5 

N.  31°  W. 

9.40       " 

Draw  NS  to  represent  a  meridian  line  ;  in  NS  take  any  con' 
venient  point,  as  A,  for  the  first  station,  and  lay  off  an  angle, 
NAB,  equal  to  45°,  the  bear- 
ing from  A  to  B,  which  will 
give  the  direction  from  A  to 
B.  Then,  from  the  scale  of 
equal  parts,  make  AB  equal 
to  9.30,  the  length  of  the  first 
side  :  this  will  give  the  sta- 
tion B.  Through  B  draw  a 
second  meridian  parallel  to 
NS  ;  lay  off  an  angle  of  60°, 
and  make  the  line  BC  equal  to 
11.85.  Proceed  in  the  same 
manner  \vith  the  other  sides.  If  the  survey  is  correct,  and  the 
plotting  accurately  performed,  the  end  of  the  last  side,  EA, 
will  fall  on  A,  the  place  of  beginning.  This  plot  is  made  on  a 
scale  of  10  chains  to  an  inch. 

(154.)  To  avoid  the  inconvenience  of  drawing  a  meridian 
through  each,  angle  of  the  field,  the  sides  may  be  laid  down 
from  the  angles  which  they  make  with  each  other,  instead  of 
the  angles  which  they  make  with  the  meridian.  Reverse  one 
of  the  bearings,  if  necessary,  so  that  both  bearings  may  run 
from  the  same  angular  point ;  then  the  angle  which  any  two 
contiguous  sides  make  writh  each  other  may  be  determined 
from  the  following 

RULES. 

1.  If  both  courses  are  north  or  south  and  both  east  or  west, 
subtract  the  less  from  the  greater. 

2.  If  both  are  north  or  south,  but  one  east  and  the  other 
west,  add  t\en:  together. 


1.06 


TRIGONOMETRY. 


3.  If  one  is  nortli  and  the  other  south,  but  both  east  or  west 
subtract  their  sum  from  180°. 

4.  If  one  is  north  and  the  other  south,  one  east  and  the  othoi 
\vest,  subtract  their  difference  from  180°. 

Thus  the  angle  CAB  is  equal  to 
\AB-NAC. 

The  angle  CAD  is  equal  to  NAG 
V-NAD. 

The  angle 'DAF  is  equal  to  180° 
-(NAD+SAF). 

The  angle  CAF  is  equal  to  180° 
(SAF-NAC). 

In  the  pieceding  example  we  ac- 
cordingly find  the  angle 

ABC=105°.  DEA-1010. 

BCD=100°.  EAB=104°. 

CDE=130°. 

With  these  angles  the  field  may  he  plotted  without  drawing 
parallels. 

EXAMPLE  2. 
The  following  field  notes  are  given  to  protract  the  survoy: 


Stations. 

Bearings. 

Distances. 

1 

N.  50°  30'  E. 

16.50  chains. 

2 

S.  68°  15'  E. 

14.20       « 

3 

S.     9°  45'  E. 

8.45       " 

4 

S.  21°    0'  W. 

6.84       « 

5 

S.  73D  30'  W. 

12.31       « 

6 

N.  78°  15'  W. 

9.76       « 

7 

N.  15°  30'  W. 

11.55       " 

THE  TRAVERSE  TABLE. 

'155.)  The  accompanying  traverse  table  shows  the  difference 
of  latitude  and  the  departure  to  four  decimal  places,  for  dis- 
tances from  1  to  10,  and  for  hearings  from  0°  to  90°,  at  inter- 
vals of  15'.  If  the  bearing  is  less  than  45°,  the  angle  will  be 
found  on  the  left  margin  of  one  of  the  pages  of  the  table,  and 
the  distance  at  the  top  or  bottom  of  the  page ;  the  difference 


SURVEYING.  107 

oi  latitude  will  be  found  in  the  column  headed  Lat.  at  the  top 
of  the  page,  and  the  departure  in  the  column  headed  Dep.  If 
the  bearing  is  more  than  45°,  the  angle  will  be  found  on  the 
right  margin,  and  the  difference  of  latitude  will  be  found  in 
the  column  marked  Lai.  at  the  bottom  of  the  page,  and  the 
departure  in  the  other  column.  The  latitudes  and  departures 
for  different  distances  with  the  same  bearing  are  proportional 
to  the  distances.  Therefore  the  distances  may  be  reckoned  as 
tens,  hundreds,  or  thousands,  if  the  place  of  the  decimal  point 
in  each  departure  and  difference  of  latitude  be  changed  ac- 
cordingly. 

Ex.  1.  To  find  the  latitude  and  departure  for  the  course  45* 
and  the  distance  93. 

Under  distance  9  on  page  141,  and  opposite  45°,  will  be 
found  latitude  6.3640  and  departure  6.3640.  Hence,  for  dis- 
tance 90,  the  latitude  is  63.640,  and  adding  the  latitude  for  the 
distance  3,  viz.,  2.121,  we  find  the  latitude  for  distance  93  to 
be  65.761. 

Ex.  2.  To  find  1^-»  latitude  and  departure  for  the  course  60a 
and  the  distance  11.35. 


Departure  for  10  is  8.6603. 

"  "      1  is  .8660. 

"  "     .8  is  .6928. 

"  "  .05  is  .0433. 


The  latitude  for  10  is  5.0000. 
"  "  "  1  is  .5000. 
u  "  "  .8  is  .4000. 
"  "  "  .05  is  .0250. 
Latitude  for  11.85  is  5.9250.  Depart,  for  11.85  is  10.2624 

Ex.  3.  To  find  the  latitude  and  departure  for  the  courso 
20°  and  the  distance  5.30. 

Ans.  Latitude  4.98,  and  departure  LSI. 

The  traverse  table  may  be  used  not  only  for  obtaining  de- 
parture and  difference  of  latitude,  but  for  finding  by  inspection 
the  sides  and  angles  of  any  right-angled  triangle  ;  for  the  lati- 
tude and  departure  form  the  two  legs  of  a  right-angled  trian- 
gle, of  which  the  distance  is  the  hypothenuse,  and  the  courso 
is  one  of  the  acute  angles. 

In  this  manner  we  find  the  latitude  and  departure  for  each 
side  df  the  field  given  in  Example  1,  page  105,  tc  be  as  in  the 
following  table : 


108 


JL  R  I  G  O  N  0  M  E  T  R  Y. 


Oc  urses. 

Dis- 
tances. 

Latitude. 

Departure. 

Cor. 
Lat. 

Cor. 
Dep. 

Balanced. 

N. 

s. 

E. 

W. 

N. 

S. 

E. 

W. 

1  N.  45°  E. 
2S.   60°  E. 
3S.  20°  W. 

9.30 
11.85 
5.30 

6.58 

5.92 

4.98 

6.58 
10.26 

1.81 

+.01 

+.01 
+.01 
—.01 

6.58 

5.93 

4.98 

6.59 
10.27 

1.80 

4  S.  70°  W. 

10.90 

3.73 

10.24 

—.01 

3.73 

10.23 

5N.  31°  W. 

940 

8.06 

4.84 

—.01 

8.06 

4.83 

Perimeter  46.75 

1464J14.63 

16.84 

16.89 

14.64 

14.64 

1686 

16.86] 

(156.)  When  a  field  has  been  correctly  surveyed,  and  the 
latitudes  and  departures  accurately  calculated,  the  sum  of  the 
northings  should  be  equal  to  the  sum  of  the  southings,  and  the 
sum  of  the  eastings  equal  to  the  sum  of  the  westings.  If  th/; 
northings  do  not  agree  with  the  southings,  and  the  eastings 
with  the  westings,  there  must  be  an  error  either  in  the  survey 
or  in  the  calculation.  In  the  preceding  example,  the  north- 
ings exceed  the  southings  by  one  link,  and  the  westings  ex- 
ceed the  eastings  by  five  links.  Small  errors  of  this  kind  are 
unavoidable;  but  when  the  error  does  not  exceed  one  link  to 
a  distance  of  three  or  four  chains,  it  is  customary  to  distribute 
the  error  among  the  sides  by  the  following  proportion : 

As  the  perimeter  of  the  field, 

Is  to  the  length  of  one  of  the  sides, 

So  is  the  error  in  latitude  or  departure, 

To  the  correction  corresponding  to  that  side, 

This  correction,  when  applied  to  a  column  in  which  the  sum 
of  the  numbers  is  too  small,  is  to  be  added ;  but  if  the  sum  of 
the  numbers  is  too  great,  it  is  to  be  subtracted. 

We  thus  obtain  the  corrections  in  columns  8  and  9  of  the 
preceding  table  ;  and  applying  these  corrections,  we  obtain  the 
balanced  latitudes  and  departures,  in  which  the  sums  of  the 
northings  and  southings  are  equal,  and  also  those  of  the  east- 
ings  and  westings. 

As  the  computations  are  generally  carried  to  but  two  deci- 
mal places,  the  corrections  of  the  latitudes  and  departures  are 
only  required  to  the  nearest  link,  and  these  corrections  may 
often  be  found  by  mere  inspection  without  stating  a  formal 
proportion.  Thus,  in  the  preceding  example,  since  the  depart- 
ures require  a  correction  of  five  links,  and  the  field  has  five 
sides  which  are  not  very  unequal,  it  is  obvious  that  we  must 
make  a  correction  of  one  liiiiv  on  each  side. 


SURVEYING. 


It  is  the  opinion  of  some  surveyors  that  when  the  error  in 
latitude  or  departure  exceeds  one  link  for  every  five  chains  of 
the  perimeter,  the  field  should  he  resurveyed  ;  hut  most  sur« 
veyors  do  not  attain  to  this  degree  of  accuracy.  The  error, 
however,  should  never  exceed  one  link  to  a  distance  of  two  01 
three  chains. 

(157.)  To  find  the  area  of  the  field. 

Let  ABODE  he  the  field 
to  he  measured.  Through  I  .„ 

A,  the  most  western  station, 
draw  the  meridian  NS,  and 
upon  it  let  fall  the  perpen- 
diculars BF,  CG,  DH,  EL 

Then  the  area  of  the  re- 
quired field  is  equal  to 
FBCDEI-(ABF+AEI). 

But  FBCDEI  is  equal  to 
the  sum  of  the  three  trape- 
zoids  FBCG-,  GCDH,  HDEI. 

Also,  if  the  sum  of  the 
parallel  sides  FB,  GO  he  multiplied  "by  FG,  it  will  give  twiou 
the  area  of  FBCG  (Art.  87).  The  sum  of  the  sides  GO,  DH, 
multiplied  hy  GH,  gives  twice  the  area  of  GCDH  ;  and  the 
sum  of  HD,  IE,  multiplied  hy  HI,  gives  twice  the  area  of 
HDEI. 

Now  BF  is  the  departure  of  the  first  side,  GO  is  the  sum  of 
(he  departures  of  the  first  and  second  sides,  HD  is  the  alge- 
braic sum  of  the  three  preceding  departures,  IE  is  the  algebraic 
sum  of  the  fr»ur  preceding  departures.  Then  the  sum  of  the 
parallel  sides  of  the  trapezoids  is  ohtained  hy  adding  together 
the  preceding  meridian  distances  two  hy  two  ;  arid  if  these 
sums  are  multiplied  hy  FG,  GH,  &o.,  which  are  the  corre- 
sponding latitudes,  it  will  give  the  'double  areas  of  the  trape- 


(158.)  It  is  most  convenient  to  leduce  all  these  operations 
a  tabular  form,  according  to  the  following 

RULE, 
Having  arranged  the  balanced  latitudes  and  departures  in 


uo 


TRIGONOMETRY. 


their  appropriate  columns,  draw  a  meridian  through  the  mosi 
eastern  or  western  station  of  the  survey,  and,  calling'  this  the 
first  station,  form  a  column  of  double  meridian  distances.  . 

The  double  meridian  distance  of  the  first  side  is  equal  ta 
its  departure  ;  and  the  double  meridian  distance  of  any  side 
is  equal  to  the  double  meridian  distance  of  the  preceding  side, 
plus  its  departure,  plus  the  departure  of  the  side  itself. 

Multiply  each  double  meridian  distance  by  its  correspond* 
ing  northing  or  southing,  and  place  the  product  in  the  column 
of  north  or  south  areas.  The  difference  between  the  sum  of 
the  north  areas  and  the  sum  of  the  south  areas  will  be  double, 
the  area  of  the  field. 

It  must  be  borne  in  mind  that  by  the  term  plus  in  this  rule 
is  to  be  understood  the  algebraic  sum.  Hence,  when  the 
double  meridian  distance  and  the  departure  are  both  east  or 
both  west,  they  must  be  added  together  ;  but  if  one  be  east 
and  the  other  west,  the  one  must  be  subtracted  from  the  other. 

The  double  meridian  distance  of  the  last  side  should  always 
be  equal  to  the  departure  for  that  side.  This  coincidence  af- 
fords a  check  against  any  mistake  in  forming  the  column  of 
double  meridian  distances. 

The  preceding  example  will  then  be  completed  as  follows  : 


N. 

s. 

i:. 

vv 

D.M.D. 

N.  Areas. 

S.  Areas.         | 

1 

2 
3 
4 
5 

6.5« 
8.06 

5.93 
4.98 
3.73 

6.59 
10.27 

1.80 
10.23 

4.83 

6.59 
23.45 
31.92 

19.89 

4.83 

43.3622 

38.9298 

139.0585 
158.9616 
74.1897 

1 

82.2920 

372.2098  1 

1 

Twice  the  figure  FBCDEI  is  372.2098  square  chains. 

Twice  the  figure  FBAEI     is    82.2920 

The  difference  is     .....  289.9178  " 

Therefore  the  area  of  the  field  is  144.9589  square  chains,  or 

14.49589  acres>  which  is  equal  to  14  acres,  1  rood,  39  perches. 
Ex.  2.  It  is  required  to  find  the  contents  of  a  tract  of  land 

of  which  the  following  are  the  field  notes : 


SURVEYING. 


11 


Sta- 
tions. 

Bearings. 

Distances. 

1 

N.  £0°  30'  E. 

16.50  chains. 

2 

S.  68°  15'  E. 

14.20      « 

3 

S.     9°  45'  E. 

8.45       < 

4 

S.  21°    0'  W. 

6.84      « 

5 

S.  73°  30'  W. 

12.31      « 

6 

N.  78°  15'  W. 

9.76      « 

7 

N.  17°    0'  W. 

11.64      « 

Calculation. 


Courses. 

Dist. 

Dif.  Lat. 

Departure. 

CO, 

Balanced. 

D.M. 
D. 

N.  Areas. 

S.  Areas. 

N. 

s. 

K.    |    VV. 

N. 
10.47 

1.97 
11.11 

s. 

K. 

\v. 

1  N.  5(P  3(y  E. 
2  S.  68°  15'  E. 
3  S.    90  45'  E. 
4  S.  21°    0'  W. 
5  S.  730  30'  W. 
6  N.  78°  15'  W. 
7N.170    VW. 

16.50 
14.20 
8.45 
6.84 
12.31 
9.76 
11.64 

10.50 

1.99 
11.13 

5.26 
8.33 
^.39 
3.50 

12.73 
13.19 
1.43 

2.45 
11.80 
9.56 
3.40 

.03 
.03 
.01 
.01 
.02 
.02 
.02 

5.29 
8.34 
6.40 
3.52 

12.70 
13.16 
1.42 

2.46 
11.82 
9.58 
3.42 

12.70 
38.56 
53.14 
52.10 
37.82 
16.42 

*, 

132.9690 

32.3474 
37.9962 

203.9824 
443.1876 
333.4400 
133.1264 

79.70,23.6-123.48  27.35 

27.21 

2:5.55 

23.55 

27.28 

27.28 

205.3126 

1113.7364 

\  Error  .14  |  Error  .14 

203.3126 

Am.,  45  A.,  2  R.,  3  P. 
Ex.  3.  Required  the  area  of  a 
tract  of  land  of  which  the  follow- 
ing are  the  field  notes  : 


2)910.4238 
455.2119 


Sta- 
tions. 

Bearings. 

Distances. 

1 

N.  58°  45'  E. 

19.84  chains. 

2 

N.  39°  30'  E. 

10.45      « 

3 

S.  45°  15'  E. 

37.26      « 

4 

S.  52°  30  W. 

21.53      « 

5 

S.  34°    0'  E. 

9.12      " 

6 

S.  66°  15'  W. 

27.69      " 

7 

N.  12°  45'  E. 

24.31      « 

8 

N.  48°  15'  W. 

24.60  '    " 

Ans.,  130  A.,  2R.,23P, 

Ex.  4.  Required  the  area  of  a  piece  of  land  from  the  follow- 
Lag  field  notes : 


TRIGONOMETRY 


[  Ptations. 

Bearings. 

Distances. 

1 

N.    5°  15'  E. 

15.17  chains. 

!  2 

3 

N.  45°  45'  E. 
N.  32°    0'  W. 

16.83      « 
14.26      " 

4 

N.  88°  30'  E. 

19.54      " 

5 

S.  28°  15'  E. 

17.92      " 

6 

S.  40°  45'  W. 

9.71      " 

7 
8 
9 

S.  31°  30'  E. 
S.  14°    O'W. 
S.  82°  45'  W. 

22.65      " 
18.39      " 
24.80      " 

10 

N.  23°  15'  W.          26.31      « 

Ans.,  173  A.,  0  R.,  23  P. 


En  fi 

.  Required  the  area  of  a  field  from  the  followin 

Stations. 

Bearings. 

Distances. 

1 

N.  32°  15'  E. 

28.74  chains. 

2 

N.  17°  45'  E. 

21.59      " 

3 

S.  81°  30'  E. 

13.38      " 

4 

S.     9°  45'  W. 

11.92      " 

5 

S.  43°    0'  E. 

19.65      " 

6 

N.  25°  30'  E. 

17.26      « 

7 

S.  78°  15'  E. 

18.87      " 

8 

S.     5°  45'  W. 

31.41      « 

9 

S.  37°  30'  W. 

26.13      « 

10 

•  N.  69°    0'  W. 

23.86      « 

11 

S.  74°  15'  W. 

20.91      « 

12 

N.  27°  30'  W. 

23.20      " 

Ans.,  304  A.,  2  R.,  9  P. 


Rv.  6.  Required  the  area  of  a  field  from  the  following 


Stations. 

Bear  in  ITS. 

Distances. 

1 

N.  36°  15'  E. 

24.73  chains. 

2 

N.    7°  45'  E. 

11.58      " 

3 

N.  79°  30'  E. 

15.39      « 

4 

S.  86°  45'  E. 

•    20.56      « 

5 

S.  12°  15'  W. 

18.14      " 

6 

S..  25°    0'  E. 

21.92      " 

7 

S.  58°  30'  W. 

29.27      " 

8 

N.  34°    0'  W. 

19.81      " 

9 

N.  81°  15'  W. 

21.24      « 

..  1.79  A.,  1  R.,  fi  f 


SURVEYING. 


113 


(159  )  The  field  notes  from  which  the  area  is  to  be  com- 
puted may  "be  imperfect.  There  may  be  obstacles  which  pre- 
vent the  measuring  of  one  side,  or  the  notes  may  be  defaced 
so  as  to  render  some  of  the  numbers  illegible.  If  the  bearing?, 
and  lengths  of  all  the  sides  of  a  field  except  one  are  given,  the 
remaining  side  may  easily  be  found  by  calculation.  For  the 
difference  between  the  sum  of  the  northings  and  the  sum  of 
the  southings  of  the  given  sides  will  be  the  northing  or  south- 
ing of  the  remaining  side  ;  and  the  difference  between  the  sum 
of  the  eastings  and  the  sum  of  the  westings  of  the  given  sides 
will  be  the  easting  or  wssting  of  the  remaining  side.  Having, 
then,  the  difference  of  latitude  and  departure  of  the  required 
side,  its  length  and  direction  are  easily  found  by  Trigonome- 
try (Art.  47). 

Ex.  Griven  the  bearings  and  lengths  of  the  sides  of  a  tract 
of  land  as  follows  : 


Stations. 

Bearings. 

Distances.                   ] 

1 

N.  18°  15'  E. 

8.93  chains. 

2 

N.  79°  45'  E. 

15.64       " 

3 

S.  25°    0'  E. 

14.27       " 

4 

Unknown. 

Unknown. 

5 

N.  87°  30'  W. 

18.52  chains. 

6 

N.  41°  15'  W. 

12.18       « 

Required  the  bearing  and  distance  of  the  fourth  side. 

Ans.,  S.  15°  33'  E.,  distance  8.62  chains. 

(160.)  There  is  another  method  of  finding  the  area  of  a  fieJd 
which  may  be  practiced  when  great  accuracy  is  not  required 
It  consists  in  first  drawing  a  plan  of  the  field,  as  in  Art.  153 
then  dividing  the  field  into 
triangles  by  diagonal  lines, 
and  measuring  the  bases  and 
perpendiculars  of  the  triangles 
upon  the  same  scale  of  equal 
parts  by  which  the  plot  was 
drawn.  Thas,  if  we  take  Ex. 
1,  and  draw  the  diagonals  AC, 
AD,  the  field  will  be  divided 
into  three  triangles,  whose  area 
e  easily  found  when  we  know 

H 


114 


TRIGONOMETRY. 


the  diagonals  AC,  AD,  and  the  perpendiculars  BF,  DO,  EIL 
The  diagonal  AC  is  found  by  measurement  upon  the  scale  of 
equal  parts  to  be  16.87 ;  the  diagonal  AD  is  15.67  ;  the  perpen- 
licular  BF  is  6.30 ;  DG-  is  4.92 ;  and  EH  is  6.42.  Hence 

the  triangle  ABC  =--16.87x3.15=  53.14 
"'        "       ADC-16.87X2.46=  41.50 
=  15.67x3.21=  50.30 


the  figure  ABCDE 


--144.94  sq.  chains. 

This  method  of  finding  the  area  of  a  field  is  very  expedi- 
tious, and  when  the  plot  is  carefully  drawn,  may  afford  results 
sufficiently  precise  for  many  purposes. 

(161.)  To  survey  an  irregular  boundary  by  means  of  off- 
sets. 

When  the  boundaries  of  a  field  are  very  irregular,  like  a 
river  or  lake  shore,  it  is  generally  best  to  run  a  straight  line, 
coming  as  near  as  is  convenient  to  the  true  boundary,  and 
measure  the  perpendicular  distances  of  the  prominent  points 
of  the  boundary  from  this  line. 

Let  ABCD  be  a  piece  of  land  to  be  surveyed  ;  the  lar^d  be- 
ing bounded  on  the  east  by  a  lake,  and  on  the  west  by  &  creek 
We  select  stations  A,  B,  C, 
D,  so  as  to  form  a  polygon 
which  shall  embrace  most  of 
the  proposed  field,  and  find 
its  area.  We  then  measure 
perpendiculars  aa',  bb',  cc', 
&c.,  as  also  the  distances  A#, 
ab,  be,  &c.  Then,  consider- 
ing the  spaces  Aaa1,  abb'a', 
&c.,  as  triangles  or  trapczoids, 
their  area  may  be  computed ; 
and,  adding  these  areas  to  the 
figure  ABCD,  we  shall  obtain 
the  area  of  the  proposed  field  nearly. 

(162.)  To  determine  the  bearing1  and  distance  from  onf 
point  to  another  by  means  of  a  series  of  triangles. 

When  it  is  required  to  find  the  distance  between  two  points 
remote  from  each  oth?r,  we  form  a  series  of  triangles  such  tha* 


SURVEYING. 


D 


the  first  and  second  triangles  may  have  one  side  in  common  ; 
the  second  and  third,  also,  one  side  in  common  ;  the  third  and 
fourth,  &c.  "vVe  then  measure  one  side  of  the  first  triangle 
for  a  base  line,  and  all  the  angles  in  each  of  the  triangles. 
These  data  are  sufficient  to  determine  the  length  of  the  sides 
of  each  triangle  ;  for  in  the  first  triangle  we  have  one  side  and 
the  angles  to  find  the  other  sides.  When  these  are  found,  we 
shall  have  one  side  and  all  the  angles  of  a  second  triangle  to 
find  the  other  sides.  In  the  same  manner  we  may  calculate 
the  dimensions  of  the  third  triangle,  the  fourth,  and  so  on.  "Wo 
shall  illustrate  this  method  by  an. example  taken  from  the 
Coast  Survey  of  the  United  States. 

The  object  here  is  to  make  a  survey  of  Chesapeake  Bay  and 
its  vicinity ;  to  determine  with  the 
utmost  precision  the  position  of  the 
most  prominent  points  of  the  country, 
to  which  subordinate  points  may  be 
referred,  and  thus  a  perfect  map  of 
the  country  be  obtained.  According- 
ly, a  level  spot  of  ground  was  select- 
ed on  the  eastern  side  of  the  bay,  on 
Kent  Island,  where  a  base  line,  AB, 
<)f  more  than  five  miles  in  length, 
was  measured  with  every  precaution. 
A  station,  C,  was  also  selected  upon 
the  other  side  of  the  bay,  near  An- 
napolis, so  situated  that  it  was  visi- 
ble from  A  and  B.  The  three  angles 
of  the  triangle  ABC  were  then  meas- 
ured with  a  large  theodolite,  after 
which  the  length  of  BC  may  be  com- 
puted. A  fourth  station,  D,  is  now  taken  on  the  western  shore 
of  the  bay,  visible  from  C  and  B,  and  all  the  angles  of  the  tri- 
angle BCD  are  measured,  when  the  line  BD  can  be  computed. 
A  fifth  station,  E,  is  now  taken  on  an  island  near  the  eastern 
shore,  visible  both  from  B  and  D,  and  all  the  angles  of  the  tri- 
angle BDE  are  measured,  when  DE  can  be  computed.  Also, 
all  the  ang.es  of  the  triangle  DEF  are  measured,  and  EF  i.t 
computed.  Then  all  the  angles  of  the  triangle  EFG-  are  meas 


1 J  b*  TRIGONO  M'E  T  R  Y. 

ured,  and  FG-  is  computed.  So,  also,  all  the  angles  of  the  tri. 
angle  FGH  are  measured,  and  GH  is  computed ;  and  thus  a 
chain  of  triangles  may  he  extended  along  the  entire  coast  of 
rhe  United  States.  To  test  the  accuracy  of  the  work,  it  is 
common  to  measure  a  side  in  one  of  the  triangles  remote  from 
the  first  "base,  and  compare  its  measured  length  with  that  de- 
duced by  computation  from  the  entire  series  of  triangles.  This 
line  is  called  a  base  of  verification.  Such  a  hase  has  been 
measured  on  Long  Island  ;  and,  indeed,  several  bases  have 
been  measured  on  different  points  of  the  coast.  These  are  all 
3onnected  by  a  triangulation,  and  thus  the  length  of  a  side  in 
any  triangle  may  be  deduced  from  more  than  one  base  line, 
and  the  agreement  of  these  results  is  a  test  of  the  accuracy  of 
the  entire  work.  Thus  the  length  of  one  of  the  sides  of  a  tri- 
angle  which  was  twelve  miles,  as  deduced  from  the  Kent  Island 
base,  differed  only  twenty  inches  from  that  derived  from  the 
Long  Island  base,  distant  two  hundred  miles. 

The  superiority  of  this  method  of  surveying  arises  from  the 
circumstance  that  it  is  necessary  to  measure  but  a  small  num- 
ber of  base  lines  along  a  coast  of  a  thousand  or  more  miles  in 
extent ;  and  for  these  the  most  favorable  ground  may  be  se- 
lected any  where  in  the  vicinity  of  the  system  of  triangles. 
All  the  other  quantities  measured  are  angles;  and  the  pre- 
cision of  these  measurements  is  not  at  all  impaired  by  the  in- 
equalities of  the  surface  of  the  ground.  Indeed,  mountainous 
countries  afford  peculiar  facilities  for  a  trigonometrical  survey, 
:since  they  present  heights  of  ground  visible  to  a  great  distance, 
and  thus  permit  the  formation  of  triangles  of  very  large  di- 
mensions. 

(163.)  To  divide  an  irregular  piece  of  land  into  any  two 
griven  parts. 

We  first  run  a  line,  by  estimation,  as  near  as  may  be  to  th*. 
required  division  line,  and  compute  the  area  thus  cut  off.  li 
this  is  found  too  large  or  too  small,  we  add  or  subtract  a  tri- 
angle, or  some  other  figure,  as  the  case  may  require.  Sup- 
pose it  is  required  to  divide  the  field  ABCDEFGHI  into  two 
equal  parts,  by  a  line  IL,  running  from  the  corner  I  to  the 
opposite  side  CD,  We  first  draw  a  line  from  I  to  D,  and  com- 
pute  the  area  of  the  part  DEFGHI ;  and,  knowing  the  area 


SURVEYING. 


ir 


of  the  entire 'field,  we  learn  the  area  which  must  be  contained 
in  the  triangle  DIL,  in  order  that 
IL  may  divide  the  field  into  two 
equal  parts.  Having  the  bearings 
and  distances  of  the  sides  DE,  EF, 
&c.,  we  can  compute  the  bearing 
and  distance  of  DI.  Thus  the  an- 
gle TDK  is  known ;  and,  having 
the  hypothenuse  ID,  we  can  com- 
pute the  length  of  the  perpendicu- 
lar IK  let  fall  on  CD.  Now  the 
base  of  a  triangle  must  be  equal 
to  its  area  divided  by  half  the  al- 
titude. Hence,  if  we  divide  the 
area  of  the  triangle  DIL  by  half  of  IK,  it  will  give  DL. 

In  a  similar  manner  we  might  proceed  if  it  was  required  to 
divide  a  tract  of  land  into  any  two  given  parts. 

Jj 

Variation  of  the  Needle. 

(164.)  The  line  indicated  by  a  magnetic  needle,  when  free- 
ly supported  and  allowed  to  come  to  a  state  of  rest,  is  called 
the  magnetic  meridian.  This  does  not  generally  coincide  with 
the  astronomical  meridian,  which  is  a  true  north  and  south 
line. 

The  angle  which  the  magnetic  meridian  makes  with  the 
true  meridian  is  called  the  variation  of  the  needle,  and  is  said 
to  be  east  or  west,  according  as  the  north  end  of  the  needle, 
points  east  or  west  of  the  north  pole  of  the  earth. 

The  variation  of  the  needle  is  different  in  different  parts  of 
the  earth.  In  some  parts  of  the  United  States  it  is  10°  west. 
and  in  others  10°  east,  while  at  other  places  the  variation  has 
every  intermediate  value.  Even  at  the  same  place,  the  varia- 
tion does  not  remain  constant  for  any  length  of  time.  Hence 
it  is  necessary  frequently  to  determine  the  amount  of  the  varia- 
tion, which  is  easily  done  when  we  know  the  position  of  tin? 
true  meridian.  The  latter  can  only  be  determined  by  astro- 
nomical observations.  The  best  method  is  by  observations  of 
the  pole  star.  If  this  star  were  exactly  at  the  pole,  it  would 
always  be  on  the  meridian ;  but,  being  at  a  distance  of  about 


118 


T  R  J  G  O  K  O  M  E  T  R  Y. 


a  degree  and  a  hall  from  the  pole,  it  revokes  about  the  polo  ia 
a  small  circle  in  a  little  less  than  21  hours.  In  about  six  hcmis 
from  its  passing  the  meridian  above  the  pole,  it  attains  its 
greatest  distance  west  of  the  meridian ;  in  about  six  hours 
more  it  is  on  the  meridian  beneath  the  pole  ;  and  in  about  six 
hours  more  it  attains  its  greatest  distance  east  of  the  meridian. 
If  the  star  can  be  observed  at  the  instant  when  it  is  on  the 
meridian,  either  above  or  below  the  pole,  a  true  north  and  south 
line  may  be  obtained. 

(165.)  The  following  table  shows  the  time  of  the  pole  star's 
passing  the  meridian  above  the  pole  for  every  fifth  day  of  the 
year : 


t.«1  TInj. 

Srti  Tfaj. 

irtfi  DH.V. 

I'ith  Day. 

21  st  Hay. 

2fYtli  17ar.      f 

Janoary   .  . 
February.  . 
March    .  .  . 
April         . 
May.. 
June    .... 
JB)V     . 

li.     in. 

6  20  P.M. 
4  J8     «• 
2  28     " 
0  20     " 
10  28  A.M. 
8  26     " 
6  28     " 

k     m. 
6    0  P.M. 

358     "• 
2"    8     "• 
07     " 
10    9  A.M. 

87" 
69" 

5  41  P.M. 
339     " 

1  49     « 
11  47  A.M. 

9  49     •* 
7  47     " 
5  49     " 

b.    m. 
5  21   P.M. 
3  19      " 

1  29     •< 
11  27  A.M. 
929     " 
7  27     " 
5  29     " 

h.     m. 
5    1  P.M. 
30" 

19" 
11    8  A.M. 
99" 

78" 

5  10     •* 

h.     ro.               f 

4  42  P.M. 

2  40     " 
0  50     " 

10  48  AJH. 
8  50     « 
6  4.8     " 
4  50     " 

August  .  .  . 
September  . 
October    .  . 
November  . 
IXeceinber  . 

4  27     « 
225     " 
0  26     « 
10  21  P.M. 
8  23     " 

47" 

2  5    " 

0    7     " 
10    1  P.M. 
g    3     " 

3  47     " 
1  45     " 
11  43  P.M. 
9  41      " 
743     « 

3  27     " 
1  26     " 
11  24  P.M. 
9  22     " 
7  24     " 

38" 
16" 
11    4  P.M. 

92" 
7    4     « 

2  48     " 
0  46     •* 
10  44  P.M. 
8  42     " 
6  44     " 

If  the  pole  star  passes  the  meridian  in  the  daytime,  it  can 
not  be  observed  without  a  good  telescope ;  but  llk  58m"  after 
the  dates  in  the  above  table,  the  star  will  be  on  the  meridian 
below  the  pole,  and  during  the  whole  year,  except  in  summer, 
the  pole  star  may  be  seen  with  the  naked  eye  on  the  merid- 
ian either  above  or  below  the  pole.  These  observations  are 
best  made  with  a  theodolite,  but  they  may  be  made  with  a 
common  compass.  At  5'1'  59m>  after  the  dates  in  the  above 
table,  the  star  will  have  attained  its  greatest  distance  west  of 
the  meridian  ;  and  5h-  59ra<  before  these  dates,  it  will  be  at  its 
greatest  distance  east  of  the  meridian.  In  summer,  therefore, 
we  may  observe  the  greatest  eastern  elongation  of  the  pole  star, 
at  which  time  the  star  is  1°  48'  east  of  the  true  meridian  fox 
all  places  in  the  neighborhood  of  New  York.  Making  this  al- 
lowance, a  true  meridian  is  easily  obtained ;  after  which,  the 
variation  of  the  needle  is  determined  by  placing  a  compass 
upon  this  line,  turning  the  sights  in  the  same  direction,  and 
noting  the  angle  shown  by  the  needle. 

The  following  table  shows  the  angle  which  the  plane  cf  tHc 


SURVEYING. 


Ill 


meridian  make 3  with  a  vertical  plane  passing  through  the  pole 
star,  when  at  its  greatest  eastern  or  western  elongation,  for 
any  latitude  from  30°  to  44°. 


Lat.  30° 


1°  37' 


Lat.  34°  I  Lat.  3G° 

1°  39' 11°  42' 


Lat.  38° 


Lat.  40° 

1°  48' 


Lat.  42° 
1°  51' 


Lat.  44° 

1°  55' 


1°  35' 

(166.)  Tha  variation  of  the  needle,  in  1870,  for  several  parts 
of  the  United  States,  was  as  follows  : 


Burlington,Yt.  . 
Boston,  Mass.  . 
Albany,  KY.  . 
New  Haven,  Ct. 
New  York  City 
Philadelphia  .  . 
Washington  City 


11°     7'  W. 
11°     0'  W 

8°  40'  W. 


8°  15' W. 
G°  50'W. 
5°  40' W. 
2°  40' W. 


Buffalo,  N.Y.     .    .  3°  54' 
Cleveland,  Ohio     .  0       0 
Detroit,  Mich. 


Charleston,  S.  C. 
Cincinnati,  Ohio 
Mobile,  Ala.  .  . 
St.  Louis,  Mo.  . 


1° 
1° 


O'E. 
35'  E. 
3°  IS'E. 
C°  44'  E. 
8°     O'E. 

Since  1870,  the  variation  in  New  England  has  increased 
about  four  minutes  annually  ;  in  New  York  and  Pennsylvania 
it  has  increased  from  threo  to  four  minutes  annually.  In  the 
Western  States  it  decreases  about  two  minutes  annually,  and 
in  the  Southern  States  it  decreases  about  two  minutes  annually. 


LEVELING. 

(167.)  Leveling-  is  the  art  of  determining  the  difference  oi 
level  between  two  or  more  places. 

The  surface  of  an  expanse  of  tranquil  water,  or  any  surface 
parallel  to  it,  is  called  a  level  surface.  Points  situated  in  a 
level  surface  are  said  to  be  on  the  same  level,  and  a  line  traced 
on  such  a  surface  is  called  a  line  of  true  level. 

On  account  of  the  globular  figure  of  the  earth,  a  level  sur- 
face is  not  a  plane  surface.  It  is  nearly  spherical ;  and  in  the 
common  operations  of  leveling  it  is  regarded  as  perfectly  so. 
Hence  every  point  of  a  level  surface  is  regarded  as  at  the  same 
d  istance  from  the  center  of  the  earth ;  and  the  difference  of 
level  of  two  places  is  the  difference  between  their  distances 
from  the  center. 

A  line  of  apparent  level  is  a  straight  line  tangent  to  the  sur- 
face of  the  earth. 

Thus,  if  AB  ^represent  the  surface  of  the  ocean,  the  two 
places  A  and  B  are  said  to  be  on  the  same  level;  but  if  All 


120 


TRIGONOMETRY 


be  drawn  tangent  to  the,  arc  AB  at  A,  then  AD  is  a  line  o 
apparent  level.  ^  T> 

This  is  the  line  which  is  indicated  by  a 
leveling  instrument  placed  at  A.  The  theod- 
olite may  be  employed  for  tracing  horizontal 
lines ;  but  if  nothing  further  were  required, 
there  would  be  no  occasion  for  graduated  cir- 
cles, and  several  parts  of  the  theodolite  might 
be  dispensed  with.  A  leveling  instrument,  therefore,  usually 
consists  of  a  large  spirit  level  attached  to  a  telescope,  mounted 
upon  a  stand  in  a  manner  similar  to  the  theodolite. 

(168.)  The  surveyor  should  also  be  provided  with  a  pair  oi 
leveling'  staves.     A  leveling  staff  consists  of  a 
rectangular  bar  of  wood  six  feet  in  length,  di-    A 
vided  to  inches  and  sometimes   tenths  of  an 
inch,  and  having  a  groove  running  its  entire 
length.     A  smaller  staff  of  the  same  length, 
called  a  slide,  also  divided  into  inches,  is  in- 
serted in  this  groove,  and  moves  freely  along  it. 

At  the  upper  end  of  the  slide  is  a  rectangu- 
lar board  called  a  vane,  AB,  about  six  inches 
wide.  The  vane  is  divided  into  four  equal  parts  by  two  lines, 
one  horizontal  and  the  other  vertical.  Two  opposite  parts  of 
the  vane  are  painted  white,  and  the  other  two  black,  in  order 
that  they  may  be  distinguished  at  a  great  distance. 

To  find  the  difference  of  level  between  any  two  points. 
(169.)  Set  up  the  leveling  staves  perpendicular  to  the  hori- 
zon, and  at  equal  distances  from  the  leveling  instrument 
Having  adjusted  the  level  by  means  of  the  proper  screws,  turn 
the  telescope  to  one  of  the  staves,  and  direct  an  assistant  to 
slide  up  the  vane  until  the  line  AB  coincides  with  the  center 
of  the  telescope,  and  note  the  height  of  this  line  from  the 
ground.  Turn  the  telescope  to  the  other  staff,  and  repeat  the 
same  operation.  Level  in  the  same  manner  from  the  second 
station  to  the  third,  from  the  third  to  the  fourth,  &c.  Then 
the  difference  between  the  sum  of  the  heights  at  the  back  sta- 
tions and  at  the  forward  stations  will  be  equal  to  the  difference 
nf  level  between  the  first  station  and  the  last. 


PURVEYING. 


121 


Jf  we  wish  to  level  fiom  A  to  E,  we  set  up  the  stages  at  a 
sonvenient  distance, 
AC,  and  midway  be- 
tween them  place  the 
level  B.  Observe 
where  the  line  of  lev- 
el, FGr,  cuts  the  rods,  and  note  the  heights  AF,  CGr.  Theii 
Difference  is  the  difference  of  level  between  the  first  and  second 
stations.  Take  up  the  level  and  place  it  at  D,  midway  be- 
tween the  rods  C  and  E,  and  observe  where  the  line  of  level, 
[II,  cuts  the  rods,  and  note  the  heights  CH,  El. 

Then  FA— CGr=the  ascent  from  A  to  C, 

and  CH-EI  =the  ascent  from  C  to  E. 

Therefore  (FA+CH)  —  (CG-+EI)=the  entire  ascent  from 
A  to  E  ;  and  in  the  same  manner  we  may  find  the  difference 
of  level  for  any  distance  ;  that  is,  the  difference  between  the 
sum  of  the  heights  at  the  back  stations  and  at  the  forward 
stations  is  equal  to  the  difference  of  level  between  the  first  sta- 
tion and  the  last. 

(170.)  The  following  is  a  copy  of  the  field  notes  for  running 
a  level  from  A  to  E  : 


Back  sights. 

Feet.    Inches. 

0       4 
5     10 


Fore  sights. 

Feet.    Inches. 

3      2 


2 

6 
11 

7 
1 


Sum  31       5  .am  20       7 

The  back  sights  being  greater  in  amount  than  the  forward 
sights,  it  is  evident  that  E  is  higher  than  A  by  10  feet  10 
inches. 

The  heights  indicated  by  the  leveling  staves  are  sometimes 
read  off  by  the  assistant,  but  it  is  better  for  the  observer  to 
read  off  the  quantities  himself  through  the  telescope  of  his 
leveling  instrument.  This  may  easily  be  done  provided  the 
graduation  of  the  staff  is  perfectly  iistinct ;  and  in  that  case  ii 


122 


TRIGONOMETRY. 


is  only  necessary  to  rely  upon  the  assistant  to  hold  the  staff 
perpendicularly.  To  enable  him  to  do  this,  a  small  plummet  is 
suspended  in  a  groove  cut  in  the  side  of  the  staff. 

(171.)  It  must  be  observed  that  the  lines  GrF,  HI  are  linei 
of  apparent  level,  and 
not  of  true  level;  nev- 
ertheless, we  shall  ob- 
tain the  true  differ- 
ence of  level  between 
A  and  E  by  this  method  if  the  leveling  instrument  is  placed 
midway  between  the  leveling  staves,  because  the  points  Gr  and 
F  will  in  that  case  be  at  equal  distances  from  the  earth's  cen  - 
ter.  If  the  level  is  not  placed  midway  between  the  staves, 
then  we  must  apply  a  correction  for  the  difference  between 
the  true  and  apparent  level. 

(172.)  To  find  the  difference  between  the  true  and  apparent 
level. 

Let  C  be  the  center  of  the  earth,  AB  a  portion  of  its  surface, 
and  AD  a  tangent  to  the  earth's  surface  at 
A ;   then  BD  is  the  difference  between  the       — —         D 
true  and  apparent  level  for  the  distance  AD. 

Now,  by  Greom.,  Prop.  11,  B.  IV., 


Hence 
and 


CD  =  VAC2+AD2, 


BD-v/AC8+AD2-BC. 
If  we  put  R  =  BC,  the  radius  of  the  earth, 


and  A=BD,  the  difference  between  the  true  and  ap- 

parent level,  we  shall  have 


that  is,  to  find  the  difference  between  the  true  and  apparent 
level  for  any  distance,  add  the  square  of  the  distance  to  the 
square  of  the  earth's  radius,  extract  the  square  root  of  thf. 
sum,  and  subtract  the  radius  of  the  earth. 

If  BD  represent  a  mountain,  or  other  elevated  object,  then 
AD  will  represent  the  distance  at  which  it  can  be  seen  in  con- 
sequence of  tho  curvature  of  the  earth. 


SURVEYING.  125 


Ex  1.  If  the  diameter  of  the  earth  be  TD12  miles,  and  if 
Mount  ^Etna  can  be  seen  at  sea  126  miles,  what  is  its  height? 

Ans.y  2  miles. 

Ex.  2.  If  a  straight  line  from  the  summit  of  Chimborazo 
touch  the  surface  of  the  ocean  at  the  distance  of  179  miles. 
what  is  the  height  of  the  mountain  ?  Ans.,  4.05  miles. 

From  the  preceding  formula  we  obtain 


that  is,  tf 

But  in  the  common  operations  of  leveling,  h  is  very  small  in 
comparison  with  the  radius  of  the  earth,  and  A2  is  very  small 
in  comparison  with  2R/L  If  we  neglect  the  term  h\  we  have 


/ 
whence  h—  ^-; 

xilt 

that  is,  the  difference  between  the  true  and  apparent  level  is 
nearly  equal  to  the  square  of  the  distance  divided  by  the  di 
ameter  of  the  earth. 

Ex.  1.  What  is  the  difference  between  the  true  and  apparent 
level  for  one  mile,  supposing  the  diameter  of  the  earth  to  be 
7912  miles?  Ans.,  8.008  inches,  or  8  inches  nearly. 

Ex.  2.  What  is  the  difference  between  the  true  and  apparent 
level  for  half  a  mile  ?  Ans.,  2  inches. 

d' 
In  the  equation  h  =  ^o",  since  2R  is  a  constant  quantity,  h 

varies  as  d2  ;  that  is,  the  difference  between  the  true  and  ap- 
parent level  varies  as  the  square  of  the  distance. 

Hence,  the  difference  for  1  mile  being  8  inches,     Ft  In 

the  difference  for  2  miles  is  8x2'=   32  inches=  2  8. 

"  "         3        "       8x3'=  72     "      =  6  0. 

"  "         4        "       Sx42=128     "      =10  8. 

5        "       8x52=200     "      =16  8. 

«         6        «       8x62=288     «      -24  0,  &CL 

Topographical  Maps. 

(173.)  It  is  sometimes  required  to  determine  and  represent 
upon  a  map  the  undulations  and  inequalities  in  the  surface  of 


124 


TRIGONOMETRY. 


a  tract  of  land,  Such  a  map  should  give  a  complete  view  oi 
the  ground,  so  as  to  afford  the  means  for  an  appropriate  loca« 
tion  of  buildings  or  extensive  works.  For  this  purpose,  \va 
suppose  the  surface  of  the  ground  to  he  intersected  by  a  num 
her  of  horizontal  planes,  at  equal  distances  from  each  othei 
The  lines  in  which  these  planes  meet  the  surface  of  the  ground, 
being  transferred  to  paper,  will  indicate  the  variations  in  the 
inclination  of  the  ground  ;  for  it  is  obvious  that  the  curves  will 
be  nearer  together  or  further  apart,  according  as  the  ascent  is 
steep  or  gentle. 

Thus,  let  ABCD  be  a  tract  of  broken  ground,  divided  by  a 
stream,  EF,  the  ascent  being  rapid  on  each  bank,  the  ground 
swelling  to  a  hill  A E 
at  Gr,  and  also  at 
H.  It  is  required 
to  represent  these 
inequalities  upon 
paper,  so  as  to 
give  an  exact  idea 
of  the  face  of  the 
ground.  The  low- 
est point  of  the 
ground  is  at  F. 
Suppose  the  tract 
to  be  intersected  C  ^  D 

by  a  horizontal  plane  four  feet  above  F,  and  let  this  plane  in- 
tersect  the  surface  of  the  ground  in  the  undulating  lines  marked 
4,  one  on  each  side  of  the  stream.  Suppose  a  second  horizon- 
tal plane  to  be  drawn  eight  feet  above  F,  and  let  it  intersect 
the  surface  of  the  ground  in  the  lines  marked  8.  Let  other 
horizontal  planes  be  drawn  at  a  distance  of  12,  16,  20,  24, 
&o.,  feet  above  the  point  F.  The  projection  of  these  lines  of 
te\el  upon  paper  shows  at  a  glance  the  outline  of  the  tract, 
We  perceive  that  on  the  right  bank  of  the  stream  the  ground 
rises  more  rapidly  on  the  upper  than  on  the  lower  portion  of 
the  map,  as  is  shown  by  the  lines  of  level  being  nearer  to  on« 
another.  On  the  right  bank  of  the  stream  the  ascent  is  unin 
terrupted  until  we  reach  Gr,  which  is  the  summit  of  the  hill. 
Beyond  G  the  ground  descends  again  toward  B.  On  Ihe  left 


oank  of  the  stream  the  ground  rises  to  H ;  but  toward  A  the 
level  line  of  12  feet  divides  into  two  branches,  and  between 
them  the  ground  is  nearly  level. 

(174.)  The  surveys  requisite  for  the  construction  of  such  a 
map  may  be  made  with  a  theodolite  or  common  level. 

The  object  is  to  trace  a  series  of  level  lines  upon  the  surface 
of  the  ground.  For  this  purpose  we  may  select  any  point  on 
the  surface  of  a  hill,  place  the  level  there,  and  run  a  level  line 
around  the  hill,  measuring  the  distances,  and  also  the  angles, 
at  every  change  of  direction.  "We  may  then  select  a  second 
point  at  any  convenient  distance  above  or  below  the  former, 
and  trace  a  second  level  line  around  the  hill,  and  so  on  for  as 
many  curves  as  may  be  thought  necessary.  Such  a  method, 
however,  would  not  always  be  most  convenient  in  practice. 

(175.)  The  following  method  may  sometimes  be  preferable: 
Set  up  the  level  on  the  summit  of  the  hill  at  Gr,  and  fix  the  vane 
on  the  leveling  staff  at  an  elevation  of  four  feet  in  addition  to 
the  height  of  the  telescope  above  the  ground.  Then  direct  an 
assistant  to  carry  the  leveling  staff,  holding  it  in  a  vertical  posi- 
tion, toward  K,  till  he  arrives  at  a  point,  as  a,  where  the  vano 
appears  to  coincide  with  the  cross  wires  of  the  telescope. 
This  will  determine  one  point  of  the  curve  line  four  feet  be- 
low  G-.  The  assistant  may  then  proceed  to  the  line  GrB,  and 
afterward  to  GrL,  moving  backward  or  forward  in  each  of  those 
directions  till  he  finds  points,  as  d  and  g-,  at  which  the  vane 
coincides  with  the  cross  wires  of  the  telescope.  The  horizonta] 
distance  between  Gr  and  a,  Gr  and  d,  Gr  and  g*,  must  then  be 
measured . 

If  the  leveling  staff  is  sufficiently  long,  the  vane  may  be 
fixed  on  it  at  the  height  of  eight  feet,  in  addition  to  the  height 
of  the  telescope  at  GT  ;  and  the  assistant,  placing  himself  in 
the  directions  GrK,  GrB,  GL,  must  move  till  the  vane  appears 
to  coincide  with  the  cross  wires  as  before.  The  horizontal  dis- 
tances  ab,  de,  gh,  must  then  be  measured,  and  stakes  driven 
into  the  ground  at  b,  e,  and  h. 

The  level  must  now  be  removed  to  b  ;  and  the  vane  being 
fixed  on  the  staff  at  a  height  equal  to  four  feet,  together  with 
the  height  of  the  instrument  from  the  ground  at  b,  the  as- 
sistant must  proceed  in  the  direction  Z>K,  and  stop  at  c 


T  R  I  G  0  N  0  M  E  T  R  Y 


the  vane  coincides  with  the  cross  wires  ;  then  thb  horizonta 
distance  jf  c  from  A.  E 

&  must  be  meas- 
ured. In  a  like 
manner,  the  op- 
erations may  be 
continued  from  b 
or  c  as  far  as  nec- 
essary toward  K ; 
then,  commenc- 
ing at  e,  and  aft- 
erward at  /*,  they 
may  be  continu- 
ed in  the  same  u  jr  D 
way  toward  B  and  L  respectively.  The  angles  which  the  di- 
rections GK,  GB,  GL  make  with  the  magnetic  meridian  being 
found  with  the  compass,  these  directions  may  be  represented 
on  paper.  Then  the  measured  distances  G#,  ab,  &c. ;  Gt/,  de, 
&c. ;  G£*,  gli,  &c.,  being  sot  off  on  those  lines  of  direction, 
ourves  drawn  through  «,  (/,  g- ;  b,  e,  h  ;  c,  /,  &,  &c.,  will  show 
the  contour  of  the  hill. 

The  map  is  shaded  so  as  to  indicate  the  hills  and  slopes  by 
drawing  fine  lines,  as  in  the  figure,  perpendicular  to  the  hori- 
zontal curves. 

(176.)  Another  method,  which  may  often  be  more  conven- 
ient than  either  of  the  preceding,  is  as  follows :  From  the  sum- 
mit  of  the  hill  measure  any  line,  as  GK,  and  at  convenient 
points  of  this  line  let  stakes  be  driven,  and  their  distances  from 
G  be  carefully  measured.  Then  determine  the  difference  of 
level  of  all  these  points  ;  and  if  the  assumed  points  do  not  fall 
upon  the  horizontal  curves  which  are  required  to  be  delineated, 
we  may,  by  supposing  the  slope  to  be  uniform  from  one  stake 
f>  another,  compute  by  a  proportion  the  points  where  the  hori- 
zontal curves  for  intervals  of  four  feet  intersect  the  line  GK. 
The  same  may  be  done  for  the  lines  GB  and  GL,  and  for  othe» 
lines,  if  they  should  be  thought  necessary. 

(177.)  If  the  surface  of  the  ground  is  gently  undulating,  it 
rnay  be  more  convenient  to  run  across  the  tract  a  number  of 
lines  parallel  to  one  another.  Drive  stakes  at  each  extremity 


SURVEYING.  12* 

oi  these  lint^,  and  also  at  all  the  points  along  them  wrier* 
there  is  any  material  change  in  the  inclination  of  the  ground, 
and  find  the  difference  of  level  between  all  these  stakes,  and 
their  distances  from  each  other.  Then,  if  we  wish  to  draw 
upon  a  map  the  level  lines  at  intervals  of  4,  6,  or  10  feet,  we 
may  compute  in  the  manner  already  explained  the  points 
where  the  horizontal  curves  intersect  each  of  the  parallel  lines. 
The  curve  lines  are  then  to  be  drawn  through  these  points,  ac 
cording  to  the  judgment  of  the  surveyor. 

(178.)  If  it  is  required  to  draw  a  profile  of  the  ground,  i  ** 
example,  from  vl  to  K,  draw  a 
straight  line,  G'K,  to  represent 
a  horizontal  line  to  which  the 
heights  are  referred,  and  set  off 
GV,  Gr'b't  G-'c',  &o.,  equal  to 
the  distances  of  the  stations  K  °'  ^'  ^  G 

from  the  beginning  of  the  line.  At  the  points  G',  a',  bf,  &c., 
erect  perpendiculars,  G'G,  a' a,  &c.,  and  make  them  equal  to 
the  heights  of  the  respective  stations.  Through  the  tops  of 
these  perpendiculars  draw  the  curved  line  GK,  and  it  will  b3 
the  profile  of  the  hill  in  the  direction  of  the  line  GK. 

On  setting-  out  Rail-way  Curves. 

(179.)  It  is  of  course  desirable  that  the  line  of  a  rail-way 
should  be  perfectly  straight  and  horizontal.  This,  however,  ia 
seldom  possible  for  any  great  distance ;  and  when  it  becomes 
necessary  to  change  the  direction  of  the  line,  it  should  be  done 
gradually  by  a  curve.  The  curve  almost  universally  employ, 
ed  for  this  purpose  is  the  arc  of  a  circle,  and  such  an  arc  may 
'^e  traced  upon  the  ground  by  either  of  the  following  methods. 

First  Method. — "When  the  center  of  the  circle  can  be  seen 
trorn  every  part  of  the  curve. 

Let  AB,  CD  be  two  straight  portions  of  a  road  which  it  is 
desired  to  connect  by  an  arc  of  a  circle.  Set  up  a  theodolite 
at  B  and  another  at  C,  and  from  each  point  range  a  line  at  right 
angles  to  the  lines  AB  and  CD  respectively ;  and  at  the  inter- 
section of  these  lines,  E,  which  will  be  the  center  of  the  circle, 
erect  a  signal  which  can  be  seen  from  any  point  between  B  and 
C.  The  stations  must  be  so  chosen  that  BF  equals  CF;  then 


128 


TRIGONOMETRY. 


4F 


on  th^se  linos  drive  stakes  at  equal  distances,  «,,  a2,  «3, 

mencing  from  the  points  B  and 

C.     If  r  represents  the  radius   A 

of  the  circle,  and  d  the  distance 

between  the  points  «,,  aa,  #3, 

&c.,  then  (Art.  172), 

will  be  the  distance  which  must 
be  set  off  from  the  first  point 
a,,  in  the  direction  fl^E,  to  ob- 
tain a  point  of  the  circular  arc. 
In  like  manner, 


will  be  the  distance  to  be  set  off  from  the  point  o2,  in  the  ai 
rection  &2E  ;  and,  generally, 


will  be  the  distance  to  be  set  off  at  the  nth  points  from  B  and 
C.  For  example,  let  r  be  one  mile,  or  5280  feet,  and  d  equa 
to  100  feet  ;  then, 


V52802+100'-52SO=.94  feet, 

will  be  the  distance  albl.     In  a  similar  manner,  we  find  at 
«3,  or  200  feet  from  B,  the  offset  will  be    3.79  feet 
o3,  or  300          "  "  "  8.52     " 

a,,  or  400  "  "  "  15.13     " 


3,or500 


fi, 


23.62 


(180.)  Second  Method. — When  A 
the  center  of  the  circle  can  not  be 
seen  from  every  part  of  the  curve, 
the  offsets  may  be  set  off  perpendic- 
ularly to  the  tangent  BF,  in  which 
case  they  must  be  computed  from 
the  formula 

For,  in  the  annexed  figure, 


that  is,   EH  =•/?=?. 


SURVEYING.  12^ 

If  r=5280  feet,  we  shall  find  the  offsets  at  intervals  of  100 
feet  to  be  a\^\~     -95  feet. 

aab2=  3.79  " 
a3b~3=  8.53  « 
«A'=  15.17  " 
a  A  =23.73  « 

For  small  distance*.,  the  offsets  will  be  given  with  sufficient 
accuracy  by  the  formula 


see  Art.  172. 

It  is  very  common  for  surveyors,  after  they  have  found  tho 
first  point,  b  ,  ,  of  the  curve,  to  join  the  points  B,  b  ,  ,  and  produce 
the  line  BZ>  l  to  the  distance  d,  and  from  the  end  of  this  line 
set  off  an  offset  to  determine  the  point  b2  ;  then,  producing  the 
line  #,&2,  set  off  a  third  offset  to  determine  the  point  &3,  and 
so  on.  The  objection  to  this  method  is,  that  any  error  com- 
mitted in  setting  out  one  of  the  points  of  the  curve  will  occa- 
sion an  error  in  every  succeeding  one.  Whenever  this  method, 
therefore,  is  employed,  it  should  be  checked  by  determining  the 
position  of  every  fourth  or  fifth  point  by  independent  compu- 
tation and  measurement. 

(181.)  Third  Method.  —  Where  the  radius  of  the  curve  ia 
MPp.ll,  nbce  a  theodolite  at  B,  and  point  its  telescope  toward 


C.  Place  another  theodolite  at  C,  and  point  its  telescope  to- 
ward  E,  the  point  of  intersection  of  the  lines  AB,  CD  produced. 
Then,  if  the  former  be  moved  through  any  number  of  degrees 
toward  a , ,  and  the  latter  the  same  number  of  degrees  toward 
a , ,  the  point  a ,  will  be  a  point  of  the  curve,  for  the  angle 
80,0  will  be  equal  to  BCD  (Geom.,  Prop.  16,  B.  III).  In  the 
same  manner,  aa,  a3,  &c.,  any  number  of  points  cf  the  curve, 
*nay  be  determined.  It  will  be  most  convenient  to  move  the 

I 


TRIGONOMETRY. 


theodolites  each  time  through  an  even  number  of  degrees,  foi 
example,  an  arc  of  two  degrees,  and  a  stake  must  be  driven  at 
each  of  the  points  of  intersection  al,  a2,  &3,  &c.  The  ac- 
curacy of  this  method  is  independent  of  any  undulations  in  the 
surface  of  the  ground,  so  that  in  a  hilly  country  this  method 
may  be  preferable  to  any  other. 


When  the  position  of  one  end  of  the  curve  is  not  absolutely 
determined,  the  engineer  may  proceed  more  rapidly.  Suppose 
it  is  required  to  trace  an  arc  of  a  circle  having  a  curvature  of 
two  degrees  for  a  hundred  feet. 

Place  a  theodolite  at  C,  the  point  where  the  curve  commen- 
ces, and  lay  off  from  the  line  CE,  toward  B,  an  angle  of  two 
degrees,  and  in  the  direction  of  the  axis  of  the  instrument  set 
off  a  distance  of  100  feet,  which  will  give  the  first  point  a1  oi 
the  curve.  Next  lay  off  from  CE  an  angle  of  four  degrees,  and 
from  a  l  set  off  a  distance  of  100  feet,  and  the  point  where  this 
line  cuts  the  axis  of  the  instrument  produced  will  be  the  sec- 
ond point  a2.  In  the  same  manner,  lay  off  from  CE  an  angle 
of  six  degrees,  and  from  a2  set  off  a  distance  of  100  feet,  and 
the  point  where  it  cuts  the  axis  of  the  instrument  produced 
will  be  the  third  point  az.  All  the  points  a, ,  a2,  #3,  etc.,  thus 
determined  lie  in  the  circumference  of  a  circle  (Geom.,  Prop. 
15,  B.  III.).  Circles  thus  drawn  are  generally  made  with  a 
curvature  of  one  or  two  degrees,  or  some  convenient  fraction 
of  a  degree,  for  every  hundred  feet.  This  method  is  very  ex 
tensively  practiced  in  the  United  States. 

Surveying1  Harbors. 

(182.)  In  surveying  a  harbor,  it  is  necessary  to  determine 
the  position  of  the  most  conspicuous  objects,  to  trace  the  out- 
line of  the  shore,  and  discover  the  depth  of  water  in  the  neigh- 
hnrlicod  of  the  channel.  A  smooth,  level  piece  of  ground  is 


SURVEYING. 


chosen,  on  which  a  base  line  of  considerable  length  is  meas- 
ured, and  station  staves  are  fixed  at  its  extremities.  "We  also 
erect  station  staves  on  all  the  prominent  points  to  be  surveyed, 
forming  a  series  of  triangles  covering  the  entire  surface  of  the 
harbor.  The  angles  of  these  triangles  are  now  measured  with 
a  theodolite,  and  their  sides  computed.  After  the  principal 
p  ints  have  been  determined,  subordinate  points  may  be  ascer- 
t  incd  by  the  compass  or  plane  table. 

Let  the  following  figure  be  a  map  of  a  harbor  to  be  survey- 


ed.  "We  select  the  most  favorable  position  for  a  base  line, 
which  is  found  to  be  on  the  right  of  the  harbor,  from  A  to  B. 
"We  also  erect  station  flags  at  the  points  C,  D,  E,  F,  and  G. 
Having  carefully  measured  the  base  line  AB,  we  measure  the 
three  angles  of  the  triangle  ABC,  which  enables  us  to  compute 
the  remaining  sides.  "We  then  measure  the  three  angles  of  the 
triangle  ACD,  and  by  means  of  the  side  AC,  just  computed, 
we  are  enabled  to  compute  AD  and  CD.  "We  then  measure 
the  three  angles  of  the  triangle  CDF,  and  by  means  of  the  side 
CD,  just  found,  we  are  enabled  to  compute  CF  and  DF.  Pro- 
ceeding in  the  same  manner  with  the  triangles  CEF,  DFGr,  we 
are  enabled,  after  measuring  the  angles,  to  compute  the  sides. 
(183.)  Having  determined  the  main  points  of  the  harbor,  we 
nay  proceed  to  a  more  detailed  survey  by  means  of  the  chain 


132 


TRIGONOMETRY. 


B 


and  compass.  If  it  is  required  to  trace  the  shore,  TICK,  wa 
commence  at  H,  and  observe  the  bearings  with  the  compass-, 
and  measure  the  distances  with  the  chain.  Where  the  shjre 
is  undulating,  it  is  most  convenient  to  run  a  straight  line  fo* 
a  considerable  distance,  and  at  frequent  intervals  measure  off- 
sets to  the  shore. 

When  a  great  many  objects  are  to  be  represented  upon  a 
map,  the  most  convenient  instrument  is 

The  Plane  Table. 

(184.)  The  plane  table 
is  a  board  about  sixteen 
inches  square,  designed 
to  receive  a  sheet  of  draw- 
ing paper,  and  has  two 
plates  of  brass  upon  op- 
posite sides,  confined  by 
screws,  for  stretching  and 
retaining  the  paper  upon 
the  board.  The  margin 
of  the  board  is  divided  to 
360  degrees  from  a  cen- 
ter C,  in  the  middle  of 
the  board,  and  these  are 
subdivided  as  minutely  as  the  size  of  the  table  will  admit.  On 
one  side  of  the  board  there  is  usually  a  diagonal  scale  of  equal 
parts.  A  compass  box  is  sometimes  attached,  which  renders 
the  plane  table  capable  of  answering  the  purpose  of  a  survey- 
or',^ compass. 

The  ruler,  A,  is  made  of  brass,  as  long  as  the  diagonal  of 
the  table,  and  about  two  inches  broad.  A  perpendicular  sight* 
vane,  B,  B,  is  fixed  to  each  extremity  of  the  ruler,  and  the  eye 
looking  through  one  of  them,  the  vertical  thread  in  the  other  is 
made  to-  bisect  any  required  distant  object. 

To  the  under  side  of  the  table,  a  center  is  attached  with  a 
ball  and  socket,  or  parallel  plate  screws,  like  those  of  the  the- 
udolite,  by  which  it  can  be  placed  upon  a  staff-head ;  and  the 
table  may  be  made  horizontal  by  meins  of  a  detached  spirit 
level 


SURVEYING  133 

(185.)  To  prepare  the  table  for  use,  it  mast  be  covered  with 
arawing  paper.  Then  set  up  the  instrument  at  one  of  the 
stations,  for  example,  B  (see  fig.  on  p.  131),  and  fix  a  needle  in 
the  table  at  the  point  on  the  paper  representing  that  station, 
and  place  the  edge  of  the  ruler  against  the  needle.  ;  Then  di- 
rect the  sights  to  the  station  A,  and  by  the  side  of  the  ruler 
draw  a  line  upon  the  paper  to  represent  the  direction  of  AB. 
Then,  with  a  pair  of  dividers,  take  from  the  scale  a  certain 
number  of  equal  parts  to  represent  the  base,  and  lay  off  this 
distance  on  the  base  line.  Having  drawn  the  base  line,  move 
the  ruler  around  the  needle,  direct  the  sights  to  any  object, 
as  L,  and  keeping  it  there,  draw  a  line  along  the  edge  of  the  ru- 
ler. Then  direct  the  sights  in  the  same  manner  to  any  other 
objects  which  are  required  to  be  sketched,  drawing  lines  in  their 
respective  directions,  taking  care  that  the  table  remains  steady 
during  the  operation. 

Now  remove  the  instrument  to  the  other  extremity  of  the 
base  A,  and  place  the  point  of  the  paper  corresponding  to  that 
extremity  directly  over  it.  Place  the  edge  of  the  ruler  on  the 
base  line,  and  turn  the  table  about  till  the  sights  are  directed 
to  the  station  B.  Then  placing  the  edge  of  the  ruler  against 
the  needle,  direct  the  sights  in  succession  to  all  the  objects  ob 
served  from  the  other  station,  drawing  lines  from  the  point  A 
in  their  several  directions.  The  intersections  of  these  lines 
with  those  drawn  from  the  point  B  will  determine  the  posi- 
tions of  the  several  objects  on  the  map. 

In  this  manner  the  plane  table  may  be  employed  for  filling 
in  the  details  of  a  map  ;  setting  it  up  at  the  most  remarkable 
spots,  and  sketching  by  the  eye  what  is  not  necessary  should 
be  more  particularly  determined,  the  paper  will  gradually  be- 
come a  representation  of  the  country  to  be  surveyed. 

To  determine  the  Depth  of  Water. 

(186.)  Let  signals  be  established  on  the  principal  shoals  ami 
along  the  edges  of  the  channel,  by  erecting  poles  or  anchoring 
buoys,  and  let  their  bearings  be  observed  from  two  stations  of 
the  survey.  Then  in  each  triangle  there  will  be  known  one 
side  and  the  angles,  from  which  the  other  sides  may  be  com- 
puted, and  their  positions  thus  become  known.  Then 


124  TRIGONOMETRY.. 

tain  the  precise  depth  of  water  at  each  of  the  buoys,  ana  pr<~ 
ceed  in  this  manner  to  determine  as  many  points  as  may  be 
thought  necessary. 

If  an  observer  is  stationed  with  a  theodolite  at  each  extremi- 
ty of  the  base  line,  we  may  dispense  with  the  erection  of  per- 
manent marks  upon  the  water.  One  observer  in  a  boat  may 
make  a  sounding  for  the  depth  of  water,  giving  a  signal  at  the 
same  instant  to  two  observers  at  the  extremities  of  the  base 
line.  The  direction  of  the  boat  being  observed  at  that  instant 
from  two  stations,  the  precise  place  of  the  boat  can  be  com- 
puted. In  this  way  soundings  may  be  made  with  great  ex- 
pedition. 

There  is  also  another  method,  still  more  expeditious,  which 
may  afford  results  sufficiently  precise  in  some  cases.  Let  a 
boat  be  rowed  uniformly  across  the  harbor  from  one  station  tc 
another,  for  example,  from  D  to  Gr  (see  fig.  on  p.  131),  and  let 
a  series  of  soundings  be  made  as  rapidly  as  possible,  and  the 
instant  of  each  sounding  be  recorded.  Then,  knowing  the  en- 
tire length  of  the  line  DGr,  and  the  time  of  rowing  over  it,  we 
may  find  by  proportion  the  approximate  position  of  the  boat  at 
each  sounding.  • 

If  the  soundings  are  made  in  tide  waters,  the  times  of  high 
water  should  be  observed,  and  the  time  of  each  sounding  bo 
recorded,  so  that  the  depth  of  water  at  high  or  low  tide  may 
be  computed.  In  the  maps  of  the  United  States  Coast  Survey, 
the  soundings  are  all  reduced  to  low-water  mark,  and  the  num- 
ber of  feet  which  the  tide  rises  or.  falls  is  noted  upon  the  map 

(187.)  The  results  of  the  soundings  may  be  delineated  upon 
u  map  in  the  same  manner  as  the  observations  of  level  on  page 
124.  We  draw  lines  joining  all  those  points  where  the  depth 
of  water  is  the  same,  for  example,  20  feet.  Such  a  line  is  seen 
to  be  an  undulating  line  running  in  the  direction  from  E  to  Gr. 
We  draw  another  line  connecting  all  those  points  where  the 
depth  of  water  is  40  feet.  This  line  runs  somewhat  to  the 
east  of  the  former  line,  but  nearly  parallel  with  it.  We  draw 
other  lines  for  depths  of  60  feet,  &e.  The  lines  being  thus 
drawn,  a  mere  glance  at  the  map  will  show  nearly  the  de|  \V 
of  water  at  any  point  of  the  harbor. 


SOOK  V. 

N  A.  V  I  G  A  T  I  6  N. 

(136.)  NAVIGATION  is  the  art  of  conducting  a  ship  at  sea 
from  cue  port  to  another. 

Tliure  are  two  methods  of  determining  the  situation  of  a 
vessel  at  sea.  The  one  consists  in  finding  by  astronomical  ob- 
servations her  latitude  and  longitude ;  the  other  consists  in 
measuring  the  ship's  course,  and  her  progress  every  day  from 
the  time  of  her  leaving  port,  from  which  her  place  may  be 
computed  by  trigonometry.  The  latter  method  is  the  one  tn 
be  now  considered. 

(189.)  The  figure  of  the  earth  is  nearly  that  of  a  sphere,  and 
in  navigation  it  is  considered  perfectly  spherical.  The  earth's 
axis  is  the  diameter  around  which  it  revolves  once  a  day. 
The  extremities  of  this  axis  are  the  terrestrial  pole*  ;  one  is 
called  the  north  pole,  and  the  other  the  south  pole. 

The  equator  is  a  great  circle  perpendicular  to  the  earth's 
axis. 

Meridians  are  great  circles  passing  through  the  poles  of  the 
earth.  Every  place  on  the  earth's  surface  has  its  own  meridian. 

(190.)  The  longitude  of  any  place  is  the  arc  of -the  equator 
intercepted  between  the  meridian  of  that  place  and  some  as- 
sumed meridian  to  which  all  others  are  referred.  In  most 
countries  of  Europe,  that  has  been  taken  as  the  standard  me- 
ridian which  passes  through  their  principal  observatory.  The 
English  reckon  longitude  from  the  Observatory  of  Greenwich ; 
and  in  the  United  States,  we  have  usually  adhered  to  the  En- 
glish custom,  though  we  believe  the  time  has  come  when  longi- 
tude  should  be  reckoned  from  the  Observatory  of  Washington. 

Longitude  is  usually  reckoned  east  and  west  of  the  first  me« 
ridian,  from  0°  to  180°. 

The  difference  of  longitude  of  two  places  is  the  arc  of  the 
equator  included  between  their  meridians.  It  is  equal  to  the 


1'JG  TRIGONOMETRY 

difference  c  f  their  'longitudes  if  they  are  on  the  same  sido  of 
the  first  meridian,  and  to  the  sum  of  their  longitudes  if  on  op. 
posite  sides. 

(1.91.)  The  latitude  of  a  place  is  the  arc  of  the  meridian  pass- 
ing  through  the  place,  which  is  comprehended  between  that 
place  and  the  equator 

Latitude  is  reckoned  north  and  south  of  the  equator,  from 
IT  to  90°. 

Parallels  of  latitude  are  the  circumferences  of  small  circles 
parallel  to  the  equator. 

The  difference  of  latitude  of  two  places  is  the  arc  of  a  me- 
ridian  included  between  the  parallels  of  latitude  passing 
through  those  places.  It  is  equal  to  the  difference  of  their 
latitudes  if  they  are  on  the  same  side  of  the  equator,  and  to 
the  sum  of  their  latitudes  if  on  opposite  sides. 

The  distance  is  the  length  of  the  line  which  a  vessel  de- 
scribes in  a  given  time. 

The  departure  of  two  places  is  the  distance  of  either  place 
from  the  meridian  of  the  other.  If  the  two  places  are  OK 
the  same  parallel,  the  departure  is  the  distance  between  the 
places.  Otherwise,  we  divide  the  distance  AB  into  portion? 
A&,  be,  cd,  &c.,  so  small  P 

that  the  curvature  of  the 
earth  may  be  neglected. 
Through  these  points 
we  draw  the  meridians 
P£,  PC,  &c.,-and  the  par- 
allels be,  cf,  &c.  Then 
the  departure  for  A.b  is 
eb,  for  be  it  is  fc  ;  and 
the  whole  departure  from  A  to  B  is  eb+fc+gd+hR ;  that  is, 
the  sum  of  the  departures  corresponding  to  the  small  portions 
into  which  the  distance  is  divided. 

Distance,  departure,  and  difference  of  latitude  are  measured 
in  nautical  miles,  one  of  which  is  the  60th  part  of  a  degree  at 
the  equator.  A  nautical  mile  is  nearly  one  sixth  greater  than 
an  English  statute  mile. 

The  course  of  a  ship  is  the  angle  which  the  ship's  pa  tli  makes 
with  the  meridian.  A  ship  is  said  to  continue  on  the  same 


NAVIGAI  ION 


course  when  she  cut3  every  meridian  which  she  crosses  at  the 
same  angle.  The  path  thus  described  is  not  a  straight  line, 
but  a  curve  called  a  rhumb-line. 

The  course  of  a  ship  is  given  by  the  mariner's  compass 

(192.)  The  mariner's  compass  consists  of  a  circular  piece  ol 
paper,  called  a  card,  attached  to  a  magnetic  needle,  which  is 
balanced  on  a  pin  so  as  to  move  freely  in  any  direction.  Di 
rectly  over  the  needle,  a  line  is  drawn  on  the  card,  one  end  ol 
which  is  marked  N,  and  the  other  S.  The  circumference  is 
divided  into  thirty-two  equal  parts  called  rhumbs  or  points* 
each  point  being  subdivided  into  four  equal  parts  called  quarter 
points. 

The  points  of  the  compass  are  designated  as  follows,  begin- 
ning at  north  and  go- 
ing east:  north,  north 
by  east,  north-north- 
east, northeast  by 
north,  northeast,  and 
so  on,  as  shown  in  the 
annexed  figure. 

The  interval  be- 
tween two  adjacent 
points  is  11°  15', 
which  is  the  eighth 
part  of  a  quadrant. 
On  the  inside  of  the 
compass-box  a  black 
line  is  drawn  perpen- 
dicular to  the  horizon,  and  the  compass  should  be  so  placea 
that  a  line  drawn  from  this  mark  through  the  center  of  the 
card  may  be  parallel  to  the  keel  of  the  ship.  The  part  of  the 
card  which  coincides  with  this  mark  will  then  show  the  point 
of  the  compass  to  which  the  keel  is  directed.  The  compass  is 
suspended  in  its  box  in  such  a  manner  as  to  maintain  a  hori- 
zontal position,  notwithstanding  the  motion  of  the  ship. 

The  following  taije  shows  the  number  of  degrees  am1   Ro- 
utes corresponding  to  each  point  of  the  compass  ; 


Y  R  I  G  O  N  O  M  E  T  K  T 


North. 

Pts. 

Pts. 

South. 

N.  by  E. 

N.byW. 

1 

11-  15' 

1 

S.byE. 

S.byW. 

N.N.E. 

N.N.W. 

2 

22°  30' 

2 

S.S.E. 

S.S.W. 

N.E.byN. 

N.W.byN. 

3 

33°  45' 

3 

S.E.byS. 

SW.byS. 

N.E. 

N.W. 

4 

45°    0' 

4 

S.E. 

S.W. 

N.E.  by  E. 

N.W.byW. 

5 

56°  15' 

5 

S.E.byE. 

S.W.byW. 

E.N.E. 

W.N.W. 

6 

67°  30' 

6 

E.S.E. 

W.S.W. 

E.byN. 

W.byN. 

7 

78°  45' 

7 

E.byS. 

W.  by  S. 

East. 

West. 

8 

90°    0' 

8 

East. 

West. 

(193.)  The  ship's  rate  of  sailing  is  measured  by  a  log-line. 
The  log-line  is  a  cord  about  300  yards  long,  which  is  wound 


round  a  reel,  one  end  being  attached  to  a  piece  of  thin  board 
called  a  log.  This  board  is  in  the  form  of  a  sector  of  a  circle, 
the  arc  of  which  is  loaded  with  lead  sufficient  to  give  the  board 
a  vertical  position  when  thrown  upon  the  water.  This  is  de- 
signed to  prevent  the  log  from  being  drawn  along  after  the 
vessel  while  the  line  is  running  off  the  reel. 

The  time  is  measured  by  a  sand-glass,  through  which  the 
sand  passes  in  half  a  minute,  or  the  120th  part  of 
an  hour. 

The  log-line  is  divided  into  equal  parts  called 
knots,  each  of  which  is  50  feet,  or  the  120th  part 
'if  a  nautical  mile.  Now,  since  a  knot  has  the 
same  ratio  to  a  nautical  mile  that  half  a  minute 
has  to  an  hour,  it  follows,  that  if  the  motion  of  a 
ship  is  uniform,  she  sails  as  many  miles  in  an  hour 
as  she  does  knots  in  half  a  minute.  If,  then,  seven  knots  are 
observed  to  run  off  in  half  a  minute,  the  ship  is  sailing  at  the 
rate  of  seven  miles  an  hour. 

PLANE  SAILING. 

(194.)  Plane  sailing  is  the  method  of  calculating  a  ship's 
place  at  sea  by  means  of  the  properties  of  a  plane  triangle. 
The  particulars  which  are  given  or  required  are  four,  viz.,  the 


N  L  V I  G  A  T I  0  N. 


139 


Hence  the  difference  of  lati- 


distance,  course,  difference  of  latitude,  and  departure.     CK 
these,  any  two  being  given,  the  others  may  be  found. 

Let  the  figure  EPQ  represent  a  portion  of  the  earth's  sur. 
face,  P  the  pole,  and  EQ, 
the  equator.  Let  AB 
be  a  rhumb-line,  or  the 
track  described  by  a  ship 
in  sailing  from  A  to  B 
on  a  uniform  course. 
Let  the  whole  distance 
be  divided  into  portions 
A&,  be,  &c.,  so  small 
that  the  curvature  of  the  earth  may  be  neglected.  Through 
the  points  of  division  draw  the  meridians  P&,  PC,  &c.,  and  the 
parallels  eb,  ft,  &c.  Then,  since  the  course  is  every  where 
the  same,  each  of  the  angles  ePJ),  fbc,  &c.,  is  equal  to  the 
course.  The  distances  Ae,  bf,  &c.,  are  the  differences  of  lati- 
tude of  A  and  £,  b  and  c,  &c.  Also,  eb,  fc,  &c.,  are  the  de- 
partures for  the  same  distances, 
tude  from  A  to  B  is  equal  to 

Ar+bf+cg+dh, 
and  the  departure  is  equal  to 
eb+fc+gd+hE. 

Construct  the  triangle  A'B'C'  so  that  A'b'e' 
shall  be  equal  to  A.be,  b'c'f  shall  be  equal  to 
bcf,  c'd'g'  equal  to  cdg,  and  d'E'h'  equal  to 
t/B/i.     Then  A'B'  represents  the  distance  sail-     -A- 
ed,  B'A'C'  the  course,  A'C'  the  difference  of  latitude,  and  B'C 
the  departure ;  that  is,  the  distance,  dif- 
ference of  latitude,  and  departure  are  cor- 
rectly represented  by  the  hypothenuse  and 
sides  of  a  right-angled  triangle,  of  which 
the  angie  opposite  to  the  departure  is  the 
course.     Of  these  four  quantities,  any  two 
being  given,  the  others  may  be  found. 

Plane  sailing  does  not  assume  the  earth's 
surface  to  be  a  plane,  and  does  not  involve 
any  error  even  in  great  distances. 


€,' 


Departure 


110  TR  IG  OJS'CMETRY. 

EXAMPLES. 

1.  A  ship  sails  from  Yera  Cruz  N.E.  by  If.  74  miles.     Re> 
quired  her  departure  and  difference  of  latitude 

According  to  the  principles  of  right-angled  triangles,  Art.  44, 
Radius  :  distance  :  :  sin.  course  :  departure. 

:  :  cos.  course  :  diff.  latituat, 
The  course  is  three  points,  or  33°  45' ;  hence  we  obtain 

Departure      =41.11  miles. 
Diff.  latitude=61.53  miles. 

2.  A  ship  sails  from  Sandy  Hook,  latitude  40°  28'  N.,  upon 
a  course  E.S.E.,  till  she  makes  a  departure  of  500  miles.    "What 
distance  has  she  sailed,  and  at  what  latitude  has  she  arrived  1 

By  Trigonometry,  Art.  44, 

Sin.  course  :  departure  :  :  radius  :  distance, 

:  :  cos.  course  :  diff.  latitude- 
Ans.  Distance        =541.20  miles. 

Diff.  latitude=207.11  miles,  or  3°  27'. 
Hence  the  latitude  at  which  she  has  arrived  is  37°  V  N. 

3.  The  bearing  of  Sandy  Hook  from  Bermuda  is  N.  42°  56' 
W.,  and  the  difference  of  latitude  486  miles.     Required  tho 
distance  and  departure. 

By  Trigonometry,  Art.  46, 

Radius  :  diff.  latitude  :  :  tang,  course  :  departure* 
:  :  sec.  course  :  distance. 
Ans.  Distance   =663.8  miles. 
Departure =452.1  miles. 

4.  A  ship  sails  from  Bermuda,  latitude  32°  22'  N.,  a  distance 
of  666  miles,  upon  a  course  between  north  and  east,  until  she 
finds  her  departure  444  miles.     What  course  has  she  sailed, 
and  what  is  her  latitude  ? 

By  Trigonometry,  Art.  44, 

Distance  :  radius  :  :  departure  :  sin.  course, 
Radius  :  distance  :  :  cos.  course  :  diff.  latitude. 

Ans.  Latitude^      40°  38'  N. 
Course    =N.  41°  49'  E 

5.  The  distance  from  Vera  Cruz,  latitude  19°  12'  N.,  to. Pen. 
sacola,  latitude  30°  19'  N.,  is  820  miles.     Required  the  beai- 
ing  and  departure. 


NAVIGATION.  Ill 

By  Trigonometry,  Art.  45, 

Distance  :  radius  :  :  diff.  latitude  :  cos.  course, 
Radius  :  distance  :  :  sin.  course  :  departure. 

Ans.  Bearing     =N.  35°  34'  E. 
Departure =476. 95  miles 

(5.  A  ship  sails  from  Sandy  Hook  upon  a  course  between 
south  and  east  to  the  parallel  of  35°,  when  her  departure  was 
"00  miles.     Required  her  course  and  distance. 
By  Trigonometry,  Art.  47, 

Diff.  latitude  :  radius  :  :  departure  :  tc^ng.  course, 
Radius  :  diff.  latitude  :  :  sec.  course  :  distance. 

Ans.  Course  S.  42°  27'  E. 
Distance  444.5  miles 

TRAVERSE  SAILING. 

(195.)  A  traverse  is  the  irregular  path  of  a  ship  when  sail- 
ing on  different  courses. 

The  object  of  traverse  sailing  is  to  reduce  a  traverse  to  a 
single  course,  when  the  distances  sailed  are  so  small  that  the 
curvature  of  the  earth  may  be  neglected.  "When  a  ship  sails 
on  different  courses,  the  difference  of  latitude  is  equal  to  the 
difference  between  the  sum  of  the  northings  and  the  sum  of  the 
southings ;  and,  neglecting  the  earth's  curvature,  the  departure 
is  equal  to  the  difference  between  the  sum  of  the  eastings  and 
the  sum  of  the  westings.  If,  then,  the  difference  of  latitude  and 
the  departure  for  each  course  be  taken  from  the  traverse  table, 
and  arranged  in  appropriate  columns,  the  difference  of  latitude 
for  the  whole  time  may  be  obtained  exactly,  and  the  departure 
nearly,  by  addition  and  subtraction ;  and  the  corresponding 
distance  and  course  may  be  determined  as  in  plane  sailing. 

EXAMPLES. 
1.  A  ship  sails  on  the  following  successive  tracks: 

1.  N.B.      23  miles. 

2.  E.S.E.    45      " 

3.  E.byK  34      " 

4.  North      29      « 

5.  K  by  W.  31      « 

6.  N.N.E.    17      « 

Find  the  course  and  distance  for  the  ivholo  traverao. 


142 


T  R  I  G  O  N  0  M  E  T  R  V. 


We  form  a  table  as  below,  entering  the  courses  from  thn 
table  of  rhumbs,  page  138,  and  then  enter  the  latitudes  and 
departures  taken  from  the  traverse  table. 

Traverse  Table. 


No.  |                Course. 
~ 

Distance. 

N. 

s. 

E. 

w. 

N.  45°  E. 

23 

16.26 

16.26 

2 

S.  67°  30'  E. 

45 

17.22 

41.57 

3 

N.  78°  45'  E. 

34 

6.63 

33.35 

4 

North. 

29 

29.00 

5 

N.  11°  15'  W. 

31 

30.40 

6.05 

6 

N.  22°  30'  E. 

17 

15.71 

6.51 

Sum  of  columns     .     .     .     98.00      17.22      97.69      6.0,5 

17.22                     6.05 

Diff.  latitude      .     .     .     . =80.78  N.    Dep. =91.64  E. 

Hence  the  course  is  found  by  plane  sailing  N.  48°  36'  E., 
Rnd  the  distance  =122.2  miles. 

The  proportions  are 

Diff.  latitude  :  radius  :  :  departure  :  tang,  course, 
Radius  :  diff.  latitude  :  :  sec.  course  :  distance. 

2.  A  ship  leaving  Sandy  Hook  makes  the  following  courses 
and  distances  : 

1.  S.E.      25  miles. 

2.  E.S.E.    32      « 

3.  East      17      " 

4.  E.byS.  51      " 

5.  South    45      " 

6.  S.byE.  63      " 

Required  her   latitude,   the    distanco    made,   and  the   direct 
flour  se. 

Ans.  Latitude=3S°  1'  N. 

Distance =193.7  miles. 

Course    =S.  40°  47'  E. 

3.  A  ship  from  Pensacola,  latitude  30°  19',  sails  on  1he  fol 
lowing  successive  coursss : 

1  South       48  miles. 

2  S.S.W.      23      " 
a.        SAY.        32      « 


NAVIGATION. 


4    S.W.  by  S.  76  miies. 

5.  West       17      " 

6.  W.S.W.     54      « 
Required  lier  latitude,  direct  course,  and  distance. 

.4ras.  Latitude-     27°  23'  N, 
Course    =  S.38°39'W. 
Distance  =225.0  miles. 

4.  A  ship  from  Bermuda,  latitude  32°  22',  sails  on  the  fol- 
lowing successive  courses : 

1.  N.B.        66  miles. 

2.  N.N.B.      14      « 

3.  N.B.  by  E.  45      " 

4.  East        21      «. 

5.  E.byN.     32      " 
Required  her  latitude,  direct  course,  and  distance. 

Ans.  Latitude^  33°  53'  N 
Course  =N.  57°  22'  E 
Distance =168.4  miles. 

(196.)  When  the  water  through  which  a  ship  is  moving  haa 
a  progressive  motion,  the  ship's  progress  is  affected  in  the  same 
manner  as  if  she  had  sailed  in  still  water,  with  an  additional 
course  and  distance  equal  to  the  direction  and  motion  of  the 
current.  \ 

Ex.  5.  If  a  ship  sail  125  miles  N.N.E.  in  a  current  which 
sets  W.  by  N.  32  miles  in  the  same  time,  required  her  true 
course  and  distance. 

Form  a  traverse  table  containing  the  course  sailed  by  the 
ship  and  the  progress  of  the  current,  and  find  the  difference 
of  latitude  and  departure.  The  resulting  course  and  distance 
is  found  as  in  the  preceding  examples. 

Traverse  Table. 


Courses. 

Distance. 

N. 

E. 

w. 

'  N.  22°  30'  E 
N.  78°  45'  W. 

125 
32 

115.49 
6.24 

47.84 

31.39 

Biff,  latitude  .     .     .  =121.73      47.84     31.39 
31.39 

Departure 


-16.45  E. 


144  TRI  fi  ON o ME TII  Y. 

Hence  the  course  is  found  by  plane  sailing  -"N.  7°  42'  E.,  anc 
the  distance =122.8  miles. 

Ex.  6.  A  ship  sails  S.  by  E.  for  two  hours  at  the  rate  of  9 
miles  an  hour ;  then  S.  by  W.  for  five  hours  at  the  rate  of  8 
miles  an  hour ;  and  during  the  whole  time  a  current  sets  W. 
by  N.  at  the  rate  of  two  and  a  half  miles  an  hour,  Required 
the  direct  course  and  distance. 

Ans.  The  course  is  S.  21 D  51'  W. 
Distance  57.6  miles. 

PARALLEL  SAILING. 

(197.)  Parallel  sailing"  is  when  a  ship  sails  exactly  cast  ui 
west,  and  therefore  remains  constantly  on  the  same  parallel 
of  latitude.  In  this  case  the  departure  is  equal  to  the  distance 
sailed,  and  the  difference  of  longitude  may  be  found  by  the  fol- 
lowing 

THEOREM. 

The  cosine  of  the  latitude  of  the  parallel  is  to  radius,  as 
the  distance  run  is  to  the  difference  of  longitude. 

Let  P  be  the  pole  of  the  earth,  C  the  center,  AB  a  portkvn 
of  the  equator,  and  DE  any  parallel  of  lati- 
tude ;  then  will  CA  be  the  radius  of  the 
equator,  and  FD  the  radius  of  the  parallel. 
Let  DE  be  the  distance  sailed  by  the  ship 
on  the  parallel  of  latitude,  then  the  difference 
of  longitude  will  be  measured  by  AB,  the 
arc  intercepted  on  the  equator  by  the  merid- 
ians passing  through  D  and  E. 

Since  AB  and  DE  correspond  to  the  equal  angles  ACB, 
DFE,  they  are  similar  arcs,  and  are  to  each  oilier  as  their 
radii.  Hence 

FD  :  CA  :  :  arc  DE  :  arc  AB. 

But  FD  is  the  sine  of  PD,  or  the  cosine  of  AD,  that  is,  the 
cosine  of  the  latitude,  and  CA  is  the  radius  of  the  sphere ; 
hence 

Cosine  of  latitude  :  R  :  :  distance  :  diff.  longitude. 

Cor.  Like  portions  of  different  parallels  of  latitude  are  if 
ftAch  other  as  tho  cosines  of  the  latitudes. 


NAVIGATION. 


115 


The  length  of  a  degree  of  longitude  in  different  parallels  may 
ce  computed  by  thL  theorem.  A  degree  of  longitude  at  tha 
equator  being  60  nautical  miles,  a  degree  in  latitude  40°  may 
be  found  by  the  proportion 

R  :  cosine  40°  :  :  60  :  45.96,  the  required  length. 

The  following  table  is  computed  in  the  same  manner. 

(198.)  Table  showing  the  length  of  a  degree  of  longitude 
(or  each  degree  of  latitude. 


Lat. 

Miles. 

Lat. 

Miles. 

Latj  Miles. 

Lat. 

Miles. 

lat.!  Miles. 

Lat.  Miles. 

1  59.99 

16 

57.68 

31 

51.43 

46 

41.68 

61 

29.09 

76 

14.52 

2 

59.96 

17 

57.38 

32 

50.88 

47 

40.92 

62 

28.17 

77 

13.50 

3 

59.92 

18 

57.06 

33 

50.32 

48 

40.15 

63 

27.24 

78 

12.47 

4 

59.85 

19 

56.73 

34 

49.74 

49 

39.36 

64 

26.30 

79 

11.45 

5 

59.77 

20 

56.38 

35 

49.15 

50 

38.57 

65 

25.36 

80 

10.42 

6 

59.67 

21  56.01 

36 

48.54 

51 

37.76 

66 

24.40 

81 

9.39 

7 

59.55 

22 

55.63 

37 

47.92 

52 

36.94 

67 

23.44 

82 

8.35 

8 

59.42 

23 

55.23 

38 

47.28 

53 

36.11 

68 

22.48 

83 

7.31 

9 

59.26 

24 

54.81 

39 

46.63 

54 

35.27 

69 

21.50 

84 

6.27 

10 

59.09 

25 

54.38 

40 

45.96 

55 

34.41 

70 

20.52 

85 

5.23 

11 

58.90 

26 

53.93 

41 

45.28 

56 

33.55 

71 

19.53 

86 

4.19 

12 

58.69 

27 

53.46 

42 

44.59 

57 

32.68 

72 

18.54 

87 

3.14 

13 

58.46 

28152.98 

43 

43.88 

58 

31.80 

73 

17.54 

88 

2.09 

14 

58.22 

29  52.48 

44 

43.16 

59 

30.90 

74 

16.54 

89 

1.05 

15 

57.96 

30  51.96 

45 

42.43 

60 

30.00 

75 

15.53 

90 

0.00 

Let  ABC  represent  a  right-angled  triangle ;  then,  by  Trig, 
onomotry,  Art.  41, 

cos.  B  :  R  :  :  AB  :  BC. 
But,  by  the  preceding  Theorem,  we  have 

cos.  lat.  :  R  :  :  depart.  :  diff.  long., 
trom  which  we  see  that  if  one  leg  of  a 
right-angled  triangle   represent  the    dis-    '        epa 
tance  run  on  any  parallel,  and  the  adjacent  acute  angle  be 
made  equal  to  the  degrees  of  latitude  of  that  parallel,  then  the 
hypothenuse  will  represent  the  difference  of  longitude. 

EXAMPLES. 

J.  A  ship  yails  from  Sandy  Hook,  latitude  40°  28'  N.,  longi- 
tude 74°  1'  W.,  618  miles  due  east.  Required  her  present 
longitude. 

Cos.  40°  28'  :  R  :  :  618  :  812'.3=13°  32',  the  dLTomuoe  of 
longitude. 

K 


146  TRIGONOMETRY. 

This,  subtracted  irom  74°  1',  leaves  60°  29'  W.,  the  longi- 
tudo  required. 

2.  A  ship  in  latitude  40°  saLs  due  east  through  nine  degrees 
of  longitude.     Required  the  distance  run. 

Ans.  413.66  miles. 

3.  A  ship  having  sailed  on  a  parallel  of  latitude  261  miles, 
finds  her  difference  of  longitude  6°  15'.     What  is  her  latitude  ? 

Ans.  Latitude  45°  54' 

4.  Two  ships  in  latitude  52°  N.,  distant  from  each  other  9* 
miles,  sail  directly  south  until  their  distance  is  150  miles 
What  latitude  do  they  arrive  at? 

Ans.  Latitude  13°  34' 

MIDDLE  LATITUDE  SAILING. 

(199.)  By  the  method  just  explained  may  be  found  the  lon- 
gitude which  a  ship  makes  while  sailing  on  a  parallel  of  lati- 
tude. When  the  course  is  oblique,  the  departure  may  be  found 
by  plane  sailing,  but  a  difficulty  is  found  in  converting  ihia 
departure  into  difference  of  longitude. 

If  a  ship  sail  from  A  to  B,  the  departure  is  equal  to  eb  +  fc 
,  which  is  less   than  AC,  but 


p 

greater  than  DB.  Navigators  have  as- 
sumed that  the  departure  was  equal  to 
the  distance  between  the  meridians  PA, 
PB,  measured  on  a  parallel  EF,  equidis- 
tant from  A  and  B,  called  the  middle  lati- 
tude. 

The  middle  latitude  is  equal  to  half  the 
sum  of  the  two  extreme  latitudes,  if  both 
are  north  or  both  south ;  but  to  half  their  difference,  if  one  is 
north  and  the  other  south. 

The  principle  assumed  in  middle  latitude  sailing  is  not  per 
fectly  correct.  For  long  distances  the  error  is  considerable; 
but  the  method  is  rendered  perfectly  accurate  by  applying  to 
the  middle  latitude  a  correction  which  is  given  in  the  accom 
panying  tables,  page  149. 

(200.)  It  has  been  shown  that  when  a  ship  sails  upon  ac 
oblique  course,  the  distance,  departure,  and  difference  of  lati 
tude  may  be  represented  by  the  sides  of  a  right-angled  trian 


NAVIGATION. 


147 


gle.  The  difference  of  longitude  is  derived  from  the  departure, 
in  the  same  manner  as  in  parallel  sailing,  the  ship  being  sup- 
posed to  sail  on  the  middle  latitude  parallel.  Hence,  if  we 
combine  the  triangle  ABC  for  plane  sailing 
with  the  triangle  BCD  for  parallel  sailing, 
we  shall  obtain  a  triangle  ABD,  by  which 
all  the  cases  of  middle  latitude  sailing  may 
be  solved. 

In  the  triangle  BCD, 

Cos.  CBD  :BC  :  :  R  :  BD  ; 
that  is,  cosine  of  middle  latitude  is  to  the 
departure,  as  radius  is  to  the  difference  of 
longitude. 

In  the  triangle  ABD,  since  the  angle  D  is 
the  complement  of  CBD,  which  represents  the  middle  latitude, 
we  have 

Sin.  D  :  AB  :  :  sin.  A  :*BD; 

that  is,  cosine  of  middle  latitude  is  to  the  distance,  as  the  sine 
of  the  course  is  to  the  difference  of  longitude. 

In  the  triangle  ABC,  we  have  the  proportion 
AC  :  BC  :  :  R  :  tang.  A. 

But  we  have  before  had  the  proportion 

Cos.  CBD  :  BC  :  :  R  :  BD. 

The  means  being  the  same  in  these  two  proportions,  we  hav* 

Cos.  CBD  :  AC  :  :  tang.  A  :  BD  ; 

that  is,  cosine  of  middle  latitude  is  to  the  difference  of  lati> 
tude,  as  the  tangent  of  the  course  is  to  the  difference  of  lon- 
gitude. 

The  middle  latitude  should  always  be  corrected  according 
to  the  table  on  page  149.  The  given  middle  latitude  is  to  be 
looked  for  either  in  the  first  or  last  vertical  column,  opposite 
to  which,  and  under  the  given  difference  of  latitude,  is  inserted 
the  proper  correction  in  minutes,  which  must  be  added  to  the 
middle  latitude  to  obtain  the  latitude  in  which  the  meridian 
distance  is  exactly  equal  to  the  departure.  Thus,  if  the  mid- 
:'le  latitude  is  41°,  and  the  difference,  of  latitude  14°,  the  cor- 
rection  will  be  found  to  be  25',  which,  added  to  the  middk 
'atituds,  gives  th<5  corrected  middle  latitude  41°  25'. 


148  TRIGONOMETRY. 

EXAMPLES. 

1.  Find  the  bearing  and  distance  of  Liverpool,  latitude  53*1 
22'  N.,  longitude  2°  52'  W.,  from  New  York,  latitude  40°  42'  NM 
longitude  74°  1'  W. 

Here  are   given  two  latitudes  and  longitudes   to  find  tht» 
course  and  distance. 

The  difference  of  latitude  is      ,     .     .     .  32°  40'  = 
The  difference  of  longitude  is   ....  71° 

The  middle  latitude  is 47° 

To  which  add  the  correction  from  p.  149 

The  corrected  middle  latitude  is    ...  47C 
Then,  according  to  the  third  of  the  preceding  theorems, 
Diff.  lat. :  cos.  mid.  lat. : :  diff.  long'. :  tang:  cour.re='N.  75°  16'  E 
To  find  the  distance  by  plaiie  sailing, 
Cos.  course  :  diff.  latitude  :  :  R  :  distance=298SA  miles 

2.  A  ship  sailed  from  Bermuda,  latitude  32°  22'  N.,  longi. 
aide  64°  38'  W.,  a  distance  of  500  miles,  upon  a  course  W.N  W 
Required  her  latitude  and  longitude  at  that  timo. 

By  plane  sailing, 

R  :  distance  :  :  cos.  course  :  diff.  latitude^  191.3. 

Therefore  the  required  latitude  is 35°  33' , 

the  middle  latitude       .......  33°  58' ; 

and  the  corrected  middle  latitude   .     .     .     .     .  33°  59 

Then  we  have 

Cos.  mid.  lat.  :  distance  :  :  sin.  course  :  diff.  long-. =557'.!. 
Therefore  the  longitude  required  is  73°  55'. 

3.  A  ship  sails  southeasterly  from  Sandy  Hook,  latitude 
40°  28'  N.,  longitude  74°  1'  W.,  a  distance  of  395  miles,  when 
her  latitude  is  34°  40'  N.     Required  her  course  and  longitude 

Ans.  Course    S.  28°  14'  E. 
Longitude  70°    5'  W. 

4.  A  ship  sails  from  Brest,  latitude  48°  23'  N.,  longitude 
4°  29'  W.,  upon  a  course  W.S.W.,  till  her  departure  is  556 
miles.     Required  the  distance  sailed  and  the  place  of  the  ship. 

Ans.  Distance  601.8  miles. 
Latitude  44°  33'  N. 
Longitude  17°  57'  'VV. 


NAVIGATION. 


MERCATOR'S  SAILING 

(201.)  Mercator's  sailing  is  a  method  of  computing  differ- 
ence of  longitude  on  the  principles  of  Mercator's  chart.  On 
this  chart,  the  meridians,  instead  of  converging  toward  the 
poles  as  they  do  on  the  globe,  are  drawn  parallel  to  each  other, 
by  which  means  the  distance  of  the  meridians  is  every  when* 


150°       12,0°       QO*        60*        30°         0" SO" 


60* 


.90° 


i5o° 


60' 

43° 
30' 
15  • 
0° 
15  ' 

30' 

45° 


75° 


made  too  great  except  at  the  equator.  To  compensate  foi 
this,  in  order  that  the  outline  of  countries  may  not  be  toe 
much  distorted,  the  degrees  of  latitude  are  proportionally  en- 
larged, so  that  the  distance  between  the  parallels  of  latitude 
increases  from  the  equator  to  the  poles.  In  latitude  60°  the 
distance  of  the  meridians  is  twice  as  great,  compared  with  a 
degree  at  the  equator,  as  it  is  upon  a  globe,  and  a  degree  of 
latitude  is  here  represented  twice  as  great  as  near  the  equator 
The  diameter  of  an  island  in  latitude  60°  is  represented  twice 
as  great  as  if  it  was  on  the  equator,  and  its  area  four  times 
too  great.  In  latitude  70°  32'  the  distance  of  the  meridians 
is  three  times  too  great,  in  latitude  75°  31'  four  times  too  great 
and  so  on,  by  which  means  the  relative  dimensions  of  coun- 
tries in  high  latitudes  is  exceedingly  distorted.  On  this  ac- 
count it  is  not  common  to  extend  the  chart  beyond  latitude  75°. 
(202.)  The  distance  of  any  parallel  upon  Mercator's  chart 
from  the  equator  has  be^n  computed,  and  is  exhibited  in  tha 


150  TRIGONOMETRY. 

accompanying  tables,  pages  142-8,  which  is  called  a  Table  of 
Meridional  Parts.  This  table  may  be  computed  in  the  fol« 
lowing  manner  : 

According  to  Art.  197,  cosine  of  latitude  is  to  radius,  as  the 
departure  is  to  the  difference  of  longitude ;  that  is,  as  a  part 
of  a  parallel  of  latitude  is  to  a  like  part  of  the  equator,  or  any 
meridian. 

But  by  Art.  28,  cosine  :  E-  :  :  R  :  secant ;  hence 

V  of  a  parallel  :  V  of  a  meridian  : :  K  :  sec.  latitude. 
But  on  Mercator's  chart  the  distance  between  the  meridians 
is  the  same  in  all  latitudes  ;  that  is,  a  minute  on  a  parallel  of 
latitude  is  equal  to  a  minute  at  the  equator,  or  a  geographical 
mile.     Hence  the  length  of  one  minute,  on  any  part  of  a  me* 
ridian,  is  equal  to  the  secant  of  the  latitude.     Thus, 
The  first  minute  of  the  meridian  =  the  secant  of  1' ; 
second         "  "         =  «  2', 

third  "  "  "  3', 

&c.,  &c. 

The  table  of  meridional  parts  is  formed  by  adding  togetlni 
the  minutes  thus  found.     Thus, 
Mer.  parts  of  l'=sec.  1' ; 
Mer.  parts  of  2'= sec.  l'+»ec.  2' ; 
Mer.  parts  of  3'— see.  I'+sec.  2'-fsec.  3' ; 
Mer.  parts  of  4'=sec.  I'+sec.  2'+sec.  3'+sec.  4', 

&c.,  &c.,  &c. 

Since  the  secants  of  small  arcs  are  nearly  equal  to  radius 
or  unity,  if  the  meridional  parts  are  only  given  to  one  tenth 
nf  a  mile,  we  shall  have 

The  meridional  parts  of  l'=1.0  mile  ; 
<:  «  «         2'=2.0     " 

«  it  a        3'=3.0     " 

"  "  "         4'=4.0     "     &o., 

as  shown  in  the  table  on  page  142. 

At  2°  33'  the  sum  of  the  small  fractions  omitted  becomes 
greater  than  half  of  one  tenth,  and  the  meridional  parts  ol 
2°  33'  is  153.1 ;  that  is,  the  meridional  parts  exceed  by  ona 
tenth  of  a  mile  the  minutes  of  latitude.  At  3°  40'  the  excess 
ia  two  tenths  of  a  mile ;  at  4°  2 17  the  excess  is  three  tenths  •- 


NAVIGATION.  151 

and  as  the  latitude  increases,  the  meridional  pares  increase 
more  rapidly,  as  is  seen  from  the  table. 

An  arc  of  Mercator's  meridian  contained  between  two  par- 
allels of  latitude  is  called  meridional  difference  of  latitude,  It 
is  found  by  subtracting  the  meridional  parts  of  the  less  latitude 
from  the  meridional  parts  of  the  greater,  if  both  are  north  or 
south,  or  by  adding  them  together  if  one  is  north  and  the  other 
south.  Thus, 

The  lat.  of  New  York  is  40°  42' ;  meridional  parts=2677.8, 
•<         New  Orleans  29°  57' :  «  "        1884.9. 

The  true  diff.  of  lat.  is  10°  45' ;  mer.  diff.  lat.  is       792.9 

If  one  latitude  and  the  meridional  difference  of  latitude  bj 
given,  the  true  difference  of  latitude  may  be  found  by  reversing 
this  process.  Thus, 

The  meridional  parts  for  New  Orleans      .     .     .     =1884.9. 

Meridional  difference  of  latitude  between  New 
York  and  New  Orleans 

Therefore  the  meridional  parts  for  New  York  =2677.8, 
/ind  the  corresponding  latitude  from  the  table  is  40°  42'. 

(203.)  If  we  take  the  figure  ABC  for 

»  -i-  ?on          i  E      Diff.  Long. 

plane  sailing,  as  on  page  139,  and  pro- 
duce AC  to  E,  making  AE  equal  to  the 
meridional  difference  of  latitude,  then 
will  DE  rp  present  the  difference  of  lon- 
gitude corresponding  to  the  departure 
BC.  FPT  we  have  seen  (Art.  202)  that 
the  dr parture  is  to  the  difference  of  lon- 
gitude as  radius  is  to  the  secant  of  lati- 
tude, which  is  also  the  ratio  of  the  true 
difference  of  latitude  to  the  meridional  difference  of  latitude. 

Now,  from  the  similarity  of  the  triangles  ABC,  ADE,  we 
have 

AC  :  AE  : :  BC  :  DE  ; 

that  is,  the  true  difference  of  latitude  is  to  the  meridional  dif> 
ference  of  latitude,  as  the  departure  is  to  the  difference  of 
longitude. 

Also,  in  'he  triangle  ADE,  we  have 

E  :  tan.  A  : :  AE  :  DE ; 


1 52  TRIGONOMETRY. 

that  is,  radius  is  to  the  tangent  of  the  course,  as  the  ?ncri<i* 
ional  difference  of  latitude  is  to  the  difference  of  .ongitude. 

EXAMPLES. 

1.  Find  the  bearing  and  distance  from  Sandy  Hook,  latitude 
403  28'  N.,  longitude  74°  V  W.,  to  Havre,  latitude  49°  29'  N., 
longitude  0°  6'  E. 

The  true  difference  of  latitude  is  9°  1'=  541'; 

meridional  difference  of  latitude  =767.1 ; 

difference  of  longitude  is  74°  7' =4447. 

Hence,  to  find  the  course  by  the  preceding  proportion, 
Mer.  diff.  lat.  :  diff.  long'.  :  :  R  :  tan.  course=~N.  80°  13'  R. 
To  find  the  distance  by  plane  sailing, 
Cos.  course  :  true  diff.  lat.  :  :  R  :  dista nee =3 183. 8  miles>. 

2.  Find  the  bearing  and  distance  from  Nantucket  Shoals,  in 
latitude  41°  4'  N.,  longitude  69°  55'  W.,  to  Cape  Clear,  in  lati- 
tude 51°  26'  N.,  longitude  9°  29'  "W. 

Ans.  Course  N.  76°  E. 

Distance  2572.9  miles 

3.  A  ship  sails  from  Sandy  Hook  a  distance  of  600  miles 
upon  a  course  S.  by  E.     Required  the  place  of  the  ship. 

The  difference  of  latitude  may  be  found  by  plane  sailing, 
the  difference  of  longitude  by  Mercator's  sailing. 

Ans.  Latitude    30°  39'.5  N. 
Longitude  71°  36'.7  W. 

4.  A  ship  sails  from  St.  Augustine,  latitude  29°  52'  N.,  lon- 
gitude 81°  25'  "W.,  upon  a  course  N.E.  by  E.,  until  her  lati- 
tude is  found  to  be  34°  40'  N.     What  is  then  her  longitude, 
and  what  distance  has  she  run  ? 

Ans.  Longitude =72°  55'  W. 
Distance    =518.4  miles. 

5.  A  ship  sails  from  Bermuda  upon  a  course  N.W.  by  W 
until  her  longitude  is  found  to  be  69°  30'  W.     What  is  then 
her  latitude,  and  what  distance  has  she  run  ? 

Ans.  Latitude  35°  4'  N. 
Distance  291.6  miles. 

6.  A  ship  sailing  from  Madeira,  latitude  32°  38'  N ,  long). 
tude  16°  f  5'  W.,  steers  westerly  until  her  latitude  i*  \ff  2  N.. 


NATIGATION.  L5£ 

and  her  departure  2425  miles.     Required  her  course,  distance 
find  longitude. 

Ans.  Course  N.  79°  37'  W 
Distance  2465.3  miles. 
Longitude  67°  9'.3  W. 

7.  Find  the  bearing  and  distance  from  Sandy  Hook,  latitude 
40°  28'  N.,  longitude  74°  1'  W.,  to  the  Cape  of  G-ood  Hopp 
latitude  34°  22'  S.,  longitude  18°  30'  E. 

Ans.  Course 
Distance 


CHARTS. 

(204.)  The  charts  commonly  used  in  navigation  are  plane 
charts,  or  Mercator^s  chart.  In  the  construction  of  the  former, 
the  portion  of  the  earth's  surface  which  is  represented  is  sup- 
posed to  be  a  plane.  The  meridians  are  drawn  parallel  to  each 
other,  and  the  lines  of  latitude  at  equal  distances.  The  dis- 
tance between  the  parallels  should  be  to  the  distance  between 
the  meridians,  as  radius  to  the  cosine  of  the  middle  latitude 
of  the  chart.  A  chart  of  moderate  extent  constructed  in  this 
manner  will  be  tolerably  correct.  The  distance  of  the  merid« 
ians  in  the  middle  of  the  chart  will  be  exact,  but  on  each  sido 
it  will  be  either  too  great  or  too  small. 

When  large  portions  of  the  earth's  surface  are  to  be  repre- 
sented, the  error  of  the  plane  chart  becomes  excessive.  To 
obviate  this  inconvenience  Merca tor's  chart  has  been  con- 
structed.  Upon  this  chart  the  meridians  are  represented  by 
parallel  lines,  and  the  distance  between  the  parallels  of  latitude 
is  proportioned  to  the  meridional  difference  cf  latitude,  as  rep- 
resented on  page  149. 

We  have  seen  that  the  meridional  difference  of  latitude  is  to 
tha  difference  of  longitude  as  radius  is  to  the  tangent  of  tho 
cour.se.  Hence,  while  the  course  remains  unchanged,  the  ratio 
of  the  meridional  difference  of  latitude  to  the  difference  of  lon- 
gitude is  constant ;  and,  therefore,  every  rhumb  line  will  bo 
represented  on  Mercator's  chart  by  a  straight  line.  TMs 
property  renders  Mercator's  chart  peculiarly  cciavwciO  to 
navigators 


1£>4  TRIGONOMETRY. 

The  preceding  sketch  affords  a  very  incomplete  view  of  the 
present  state  of  the  science  of  navigation.  The  most  accurate 
method  of  ascertaining  the  situation  of  a  vessel  at  sea  is  by 
means  cf  astronomical  observations.  For  these,  however,  the 
t  must  be  referred  to  some  treatise  on  Astronomy. 


BOOK  VI. 

SPHERICAL  TRIGONOMETRY. 

(205.)  SPHERICAL  trigonometry  teaches  how  to  determino 
the  several  parts  of  a  spherical  triangle  from  having  certain 
parts  given. 

A  spherical  triangle  is  a  portion  of  the  surface  of  a  sphere, 
bounded  hy  three  arcs  of  great  circles,  each  of  which  is  less 
than  a  semicircumference. 

RIGHT-ANGLED  SPHERICAL  TRIANGLES. 
THEOREM  I. 

(206.)  In  any  right-angled  spherical  triangle,  the  sine  of 
the  hypothenuse  is  to  radius,  as  the  sine  of  either  side  is  to 
the  sine  of  the  opposite  angle. 

Let  ABC  be  a  spherical  triangle,  right-angled  at  A ;  then 
will  the  sine  of  the  hypothenuse  BC 
be  to  radius,  as  the  sine  of  the  side 
AC  is  to  the  sine  of  the  angle  ABC. 

Let  D  be  the  center  of  the  sphere ; 
join  AD,  BD,  CD,  and  draw  CE  per- 
pendicular to  DB,  which  will,  there- 
fore, be  the  sine  of  the  hypothenuse 
BC.  From  the  point  E  draw  the 
straight  line  EF,  in  the  plane  ABD,  perpendicular  to  BD,  and 
join  CF.  Then,  because  DB  is  perpendicular  to  the  two  lines 
CE,  EF,  it  is  perpendicular  to  the  plane  CEF;  and,  conse 
quently,  the  plane  CEF  is  perpendicular  to  the  plane  ABD 
(Geom.,  Prop.  6,  B.  VII.).  But  the  plane  CAD  is  also  per- 
pendicular  to  the  plane  ABD  ;  therefore  their  line  of  common 
section,  CF,  is  perpendicular  to  the  plane  ABD ;  hence  CFD, 
CFE  are  right  angles,  and  CF  is  the  sine  of  the  arc  AC. 

Now,  in  the  right-angled  plane  triangle  CFE, 
CE  :  radius    :  CF  :  sine  CEF 


156  TRIGONOMETRY. 

But  since  CE  and  FE  are  both  at  right  angles  to  DB,  the 
angle  CEF  is  equal  to  the  inclination  of  the  planes  CBD,  ABD; 
that  is,  to  the  spherical  angle  ABC.     Therefore, 
sine  BC  :  R  :  :  sine  AC  :  sine  ABC. 

(207.)  Cor.  1.  In  any  right-angled  spherical  triangle,  tk* 
sines  of  the  sides  are  as  the  sines  of  the  opposite  angles. 

For,  by  the  preceding  theorem, 

sine  BC  :  R  :  :  sine  AC  :  sine  ABC, 
and  sine  BC  :  R  :  :  sine  AB  :  sine  ACB ; 

therefore,  sine  AC  :  sine  AB  :  :  sine  ABC  :  sine  ACB. 

Cor.  2.  In  any  right-angled  spherical  triangle,  the  cosine  of 
zither  of  the  sides  is  to  radius,  as  the  cosine  of  the  hypothenusc 
is  to  the  cosine  of  the  other  side. 

Let  ABC  be  a  spherical  triangle,  right-angled  at  A.  Do- 
scribe  the  circle  DE,  of  which  B  is 
the  pole,  and  let  it  meet  the  three 
sides  of  the  triangle  ABC  produced  in 
D,  E,  and  F.  Then,  because  BD  and 
BE  are  quadrants,  the  arc  DF  is  per- 
pendicular to  BD.  And  since  BAC 
is  a  right  angle,  the  arc  AF  is  per- 
pendicular to  BD.  Hence  the  point 
F,  where  the  arcs  FD,  FA  intersect 
each  other,  is  the  pole  of  the  arc  BD  (Geom..  Prop.  5,  Cor.  2, 
B.  IX.),  and  the  arcs  FA,  FD  are  quadrants. 

Now,  in  the  triangle  CEF,  right-angled  at  the  point  E,  ac. 
cording  to  the  preceding  theorem,  we  have 

sine  CF  :  R  :  :  sine  CE  :  sine  CFE. 

But  CF  is  the  complement  of  AC,  CE  is  the  complement  of 
BC,  and  the  angle  CFE  is  measured  by  the  arc  AD,  which  is 
the  complement  of  AB.  Therefore,  in  the  triangle  ABC,  we 
have 

cos.  AC  :  R  :  :  cos.  BC  :  cos.  AB. 

Cor.  3.  In  any  right-angled  spherical  triangle,  the  cosine 
of  either  of  the  sides  is  to  radius,  as  the  cosine  of  the  angle 
opposite  to  that  side  is  to  the  sine  of  the  other  angle. 

For,  in  the  triangle  CEF,  we  have 

sine  CF  :  R  :  :  sine  EF  :  sine  ECF. 

But  sine  CF  is  rqual  to  cos.  CA.     EF  is  the  complement  at 


SPHERICAL   TRIGONOMETRY.  1^7 

ED,  which  measures  the  angle  ABC,  that  is,  sine  EF  is  equa, 
to  cos.  ABC,  and  sine  ECF  is  the  same  as  sine  ACB ;  there- 
fore, 

cos.  AC  :  R  :  :  cos.  ABC  :  sine  ACB 

THEOREM  II. 

(208.)  In  any  right-angled  spherical  triangle,  the  sine  oj 
either  of  the  sides  about  the  right  angle  is  to  the  cotangent 
of  the  adjacent  angle,  as  the  tangent  of  the  remaining  side 
is  to  radius. 

Let  ABC  "be  a  spherical  triangle,  right-angled  at  A ;  then 
will  the  sine  of  the  side  AB  be  to  the 
cotangent  of  the  angle  ABC,  as  the 
tangent  of  the  side  AC  is  to  radius. 

Let  D  he  the  center  of  the  sphere ; 
join  AD,  BD,  CD  ;  draw  AE  perpen- 
dicular to  BD,  which  will,  therefore, 
be  the  sine  of  the  arc  AB.  Also,  from 
the  point  E  in  the  plane  BDC,  draw 
the  straight  line  EF  perpendicular  to 
BD,  meeting  DC  produced  in  F,  and 
join  AF.  Then  will  AF  "be  perpendicular  to  the  plane  ABD, 
because,  as  was  shown  in  the  preceding  theorem,  it  is  the  com- 
mon section  of  the  two  planes  ADF,  AEF,  each  perpendiculai 
to  the  plane  ADB.  Therefore  FAD,  FAE  are  right  angles, 
and  AF  is  the  tangent  of  the  arc  AC. 

"Now,  in  the  triangle  AEF,  right-angled  at  A,  we  have 
AE  :  radius  :  :  AF  :  tang.  AEF. 

But  AE  is  the  sine  of  the  arc  AB,  AF  is  the  tangent  of  ti,  -, 
arc  AC,  and  the  angle  AEF  is  equal  to  the  inclination  of  tH 
planes  CBD,  ABD,  or  to  the  spherical  angle  ABC  ;  hence 
sine  AB  :  R  :  :  tang.  AC  :  tang.  ABC. 

And  "because,  Art.  28, 

R  :  cot.  ABC  :  :  tang.  ABC  :  R  ; 
therefore,  sine  AB  :  cot.  ABC  :  :  tang.  AC     :  R. 

(209.)  Cor.  1.  In  any  right-angled  spherical  triangle,  thi 
cosine  of  the  hypothenuse  is  to  the  cotangent  of  cither  of  ikt 
oblique  angles,  as  the  cotangent  of  the  other  oblique  ang  •'•  i.« 
to  radius 


lf>8  TRIGONOMETRY. 

Let  ABC  be  a  spherical  triangle,  right-angled  at  A. 
scribe  the  circle  DEF,  of  which  B 
is  the  pole,  and  construct  the  com- 
plemental  triangle  CEF,  as  in  Cor. 
2,  Theorem  I. 

Then,  in  the  triangle  CEF,  ac- 
cording to  the  preceding  theorem,  we 
tiave 
sine  CE  :  cot.  ECF  : :  tan.  EF  :  R. 

But  CE  is  the  complement  of  BC, 
EF  is  the  complement  of  ED,  the  measure  of  the  angle  ABC ; 
and  the  angle  ECF  is  equal  to  ACB,  being  its  vertical  angle; 
hence 

cos.  BC  :  cot.  ACB  :  :  cot.  ABC  :  R. 

Cor.  2.  In  any  right-angled  spherical  triangle,  the  cosine  of 
either  of  the  oblique  angles  is  to  the  tangent  of  the  adjacent 
side,  as  the  cotangent  of  the  hypothenuse  is  to  radius. 

For,  in  the  complemental  triangle  CEF,  according  to  tho 
preceding  theorem,  we  have 

sine  EF  :  cot.  CFE  :  :  tan.  CE  :  R; 
hence,  in  the  triangle  ABC, 

cos.  ABC  :  tan.  AB  :  :  cot.  BC  :  R. 

Napier' s  Rule  of  the  Circular  Parts. 

(210.)  The  two  preceding  theorems,  with  their  corollaries, 
are  sufficient  for  the  solution  of  all  cases  of  right-angled  spheri- 
cal triangles,  and  a  rule  was  invented  by  Napier  by  means  of 
which  these  principles  are  easily  retained  in  mind. 

If,  in  a  right-angled  spherical  triangle,  we  set  aside  the  right 
angle,  and  consider  only  the  five  remaining  parts  of  the  trian- 
gle, viz.,  the  three  sides  and  the  two  oblique  angles,  then  the 
two  sides  which  contain  the  right  angle,  and  the  complements 
of  the  other  three,  viz.,  of  the  two  angles  and  the  hypothenuse, 
are  called  the  circular  parts. 

Thus,  in  the  triangle  ABC,  right-angled  at  A,  the  circular 
parts  are  AB,  AC,  with  the  complements  of  B,  BC,  and  C. 

When,  of  the  five  circular  parts,  any  on'e  is  taken  for  the 
middle  part,  then,  of  the  remaining  four,  the  two  which  are 
immediately  adjacent  to  it  on  the  right  and  left  are  called  the 


SPHERICAL   TRIGONOMETRY.  159 

adjacent  parts  ;  and  the  other  two,  each  of  which  is  separated 
from  the  middle  by  an  adjacent  part, 
are  called  opposite  parts. 

In  every  question  proposed  for  solu- 
tion, three  of  the  circular  parts  are 
concerned,  two  of  which  are  given, 
and  one  required  ;  and  of  these  three, 
the  middle  part  must  be  such  that 
the  other  two  may  be  equidistant  from  it;  that  is,  may  be 
either  both  adjacent  or  both  opposite  parts.  The  value  of  the 
part  required  may  then  be  found  by  the  following 

RULE  OF  NAPIER. 

(211.)  The  product  of  the  radius  and  the  sine  of  the  middle 
part,  is  equal  to  the  product  of  the  tangents  of  the  adjacent 
parts,  or  to  the  product  of  the  cosines  of  the  opposite  parts. 

It  will  assist  the  learner  in  remembering  this  rule  to  remark, 
that  the  first  syllable  of  each  of  the  words  tangent  and  adja- 
cent contains  the  same  vowel  a,  and  the  first  syllable  of  tho 
words  cosine  and  opposite  contains  the  same  vowel  o. 

It  is  obvious  that  the  cosine  of  the  complement  of  an  angle 
is  the  sine  of  that  angle,- and  the  tangent  of  a  complement  is 
a  cotangent,  and  vice  versa. 

In  the  triangle  ABC,  if  we  take  the  side  b  as  the  middle 
part,  then  the  side  c  and  the  complement  of  the  angle  C  are 
the  adjacent  parts,  and  the  complements  of  the  angle  B  and  of 
the  hypothenuse  a  are  the  opposite  parts.  Then,  according  to 
Napier's  rule,  R  sin.  £=tan.  c  cot.  C, 

which  corresponds  with  Theorem  II. 
Also,  by  Napier's  rule, 

R  sin.  &=sin.  a  sin.  B, 
which  corresponds  with  Theorem  I. 

Making  each  of  the  circular  parts  in  succession  tke  muldla 
part,  we  obtain  the  ten  following  equations  : 

R  sin.  #=sin.  a  sin.  B=tan.  c  cot.  C. 
R  sin.  c=sin.  a  sin.  C=tan.  b  cot.  B. 
R  cos.  B= cos.  b  sin.  C  =  cot.  a  tan.  c. 
R  cos.  a  =co8.  b  cos.  c  — cot.  B  cot.  C. 
R  cos.  C  =  cos.  c  sin.  B  =  oot.  a  tan.  ft. 


tGO  TRIGONOMETRY. 

(212.)  In  order  to  determine  whether  the  quantity  sought 
is  less  01  greater  than  90°,  the  algebraic  sign  of  each  term 
should  be  preserved  whenever  one  of  them  is  negative.  If  the 
quantity  sought  is  determined  by  means  of  its  cosine,  tangent, 
or  cotangent,  the  algebraic  sign  of  the  result  will  show  whether 
this  quantity  is  less  or  greater  than  90° ; .  for  the  cosines,  tan- 
gents, and  cotangents  are  positive  in  the  first  quadrant,  and 
negative  in  the  second.  But  since  the  sines  are  positive  in  both 
the  first  and  second  quadrants,  when  a  quantity  is  determined 
by  means  of  its  sine,  this  rule  will  leave  it  ambiguous  whether 
ihe  quantity  is  less  or  greater  than  90°.  The  ambiguity  may. 
however,  generally  be  removed  by  the  following  rule. 

In  every  right-angled  spherical  triangle,  an  oblique  angle 
and  its  opposite  side  are  ahvays  of  the  same  species ;  that  is, 
both  are  greater,  or  "both  less  than  90°. 

This  follow?  from  the. equation 

R  sin.  &=tan.  c  cot.  C ; 

where,  sirr//  sin.  b  is  always  positive,  tan.  c  must  always  have 
the  same  sign  as  cot.  C  ;  that  is,  the  side  c  and  the  opposite 
angle  C  both  belong  to  the  same  quadrant. 

(213.)  When  the  given  parts  are  a  side  and  its  opposite  an- 
gle,  the  problem  admits  of  two  solutions  ;  for  two  right-angled 
spherical  triangles  may  always  be  found,  having  a  side  and  its 
opposite  angle  the  sama'  in  both,  but  of  which  the  remaining 
sides  and  the  remaining  angle  of  the  one  are  the  supplements 
of  the  remaining  sides  and  the  remaining  angle  of  the  other. 
Thus,  let  BCD,  BAD  be  the  halves  of  two  great  circles,  and 
let  the  arc  CA  be  drawn  perpendicu- 
lar to  BD  ;  then  ABC,  ADC  are  two 
right-angled  triangles,  having  the  side  B 
AC  common,  and  the  opposite  angle 
B  equal  to  the  angle  D;  but  the  side  DC  is  the  supplement  ol 
BC,  AD  is  the  supplement  of  AB,  and  the  angle  ACD  is  trie 
supplement  of  ACB. 

EXAMPLES. 

1,  In  the  right-angled  spherical  triangle  ABC,  there  are 
given  a  =  63°  56'  and  &  — 40°.  Required  the  other  sile  c,  anr' 
the  angle?  B  and  C. 


{SPHERICAL    TRIGONOMETRY.  161 

Gild  the  side  c 

the  circular  parts  concerned  are 
the  two  legs  and  the  complement  of  the 
hypothenase;  and  it  is  evident  that  if 
the  complement  of  a  be  made  the  mid- 
dle part,  b  and  c  will  be  opposite  parts  ; 
hence,  by  Napier's  rule, 

R  cos.  a—cos,  b  cos.  c  ; 
or,  reducing  this  equation  to  a  proportion, 

cos.  b  :  R  :  :  cos.  a  :  cos.  c=54°  59'  49". 

To  find  the  angle  B. 

Here  b  is  the  middle  part,  and  the  complements  of  B  and  u 
are  opposite  parts  ;  hence 

R  sin.  b=cos.  (comp.  a)Xcos.  (comp.  B)=sin.  a  sin.  B, 
or  sin.  a  :  R  :  :  sin.  b  :  sin.  B=45°  41'  25". 

B  is  known  to  be  an  acute  angle,  because  its  opposite  side  is 
less  than  90°. 

To  find  the  angle  C. 

Here  the  complement  of  C  is  the  middle  part;  also  b  and 
the  complement  of  a  are  adjacent  parts  ;  hence 

R  cos.  C=cot.  a  tan.  b, 
or  R  :  tan.  b  :  :  cot.  a  :  cos.  0=65°  45'  57". 

Ex.  2.  In  a  right-angled  triangle  ABC,  there  are  given  tho 
hypothenuse  a=91°  42',  and  the  angle  B— 95°  6'.  Required 
the  remaining  parts. 

To  find  the  angle  C. 

Make  the  complement  of  the  hypothenuse  the  middle  part; 
Jien  R  cos.  a=cot.  B  cot.  C. 

Whence  0  =  71°  36'  47". 

To  find  the  side  c. 

Make  the  complement  of  the  angle  B  the  middle  part ;  an 4 
we  have  R  cos.  B=cot.  a  tan.  c. 

Whence  c=71°  32'  14'. 

To  find  the  side  b. 

Make  the  side  b  the  middle  part ;  then 

L 


162  TRIGONOMETRY. 

R  sin.  b=sin.  a  sin.  B. 
Whence  £=95°  22'  30". 

b  is  known  to  be  greater  than  a  quadrant,  Lecause  its  opposite 
angle  is  obtuse. 

Ex.  3.  In  the  right-angled  triangle  ABC,  the  side  b  is  20° 
4',  and  its  opposite  angle  B  36°.  Required  the  remaining 
parts. 

(  a  =48°  22'  52",  or  131°  37'    8" 

Am.  \  c  -42°  19'  17",  or  137°  40'  43". 

(  C=64°  14'  26",  or  115°  45'  34". 

This  example,  it  will  be  seen,  admits  of  two  solutions,  con- 
formably to  Art.  213. 

Ex.  4.  In  the  right-angled  spherical  triangle  ABC,  there  are 
given  the  side  c,  54°  30',  and  its  adjacent  angle  B,  44°  50'. 
Required  the  remaining  parts. 

(  C=65°  49'  53" 

Ans.  )  a  =63°  10'    4'. 

(b  =38°  59'  11" 

Why  is  not  the  result  ambiguous  in  this  case  ? 
Ex.  5.  In  the  right-angled  spherical  triangle  ABC,  the  side 
b  is  55°  28',  and  the  side  c  63°  15'.     Required  the  remaining 
narts. 

(a  =7  5°  13'    2". 
Ans.  )  B=58°  25'  47 

(  C  =  67°27'    1'. 

Ex.  6.  In  the  right-angled  spherical  triangle  ABC,  there  are 
given  the  angle  B=69°  20',  and  the  angle  C=58°  16'.  Re- 
quired the  remaining  parts. 

/  a=76°  30'  37". 

Ans.  ]b  =  65°  28'  58". 

(c  =55°  47'  46". 

(214.)  A  triangle,  in  which  one  of  the  sides  is  equal  to  a 
quadrant,  may  be  solved  upon  the  same  principles  as  right- 
angled  triangles,  for  its  polar  triangle  will  contain  a  right  <jn- 
gle.  See  Geom.,  Prop.  9,  B.  IX. 

Ex.  7.  In  the  spherical  triangle  ABC,  the  side  BC=90°,  the 
angle  C=42°  10',  and  the  angle  A=115°  20'.  Required  the 
remaining  parts. 

Taking  the  supplements  of  the  given  parts,  we  shall  hava 


SPHERICAL    TRIGONOMETRY  1G3 

in  the  polar  triangle  the  hypothenuse  a' = ISO3 -r  115°  20' ^64° 
40',  and  one  of  the  sides,  c'=180°-42°  10'=137°  50',  from 
which,  by  Napier's  rule,  we  find 

B'=115°  23' 20". 
C'=132°  2'  13". 
b'  =125°  15'  36". 

Hence,  taking  the  supplements  of  these  arcs,  we  find  the 
parts  of  the  required  triangle  are 

AC=64°  36' 40". 
AB=47°  57'  47". 
B  -54°  44' 24". 

Ex.  8.  In  the  spherical  triangle  ABC,  the  side  AC -90°,  th« 
angle  C=69°  13'  46",  and  the  angle  A=72°  12'  4".  Required 
the  remaining  parts. 

(AB=70°    8' 39". 
Am.  }BC=73°  17'  29''. 
(  B    -96°  13'  23 

OBLIQUE-ANGLED  SPHERICAL  TRIANGLES. 

THEOREM  III. 

(215.)  In  any  spherical  triangle,  the  sines  of  the  sides  are 
proportional  to  the  sines  of  the  opposite  angles. 

In  the  case  of  right-angled  spherical  triangles,  this  proposi- 
tion has  already  been  demonstrated. 
Let,  then,  ABC  be  an  oblique-angled 
triangle  ;  we  are  to  prove  that 
sin.  BC  :  sin.  AC  :  :  sin.  A  :  sin.  B. 
Through  the  point  C  draw  an  arc 
of  a  great  circle  CD  perpendicular  to      ^ 
AB.     Then,  in  the  spherical  triangle 
A  CD,  right-angled  at  D,  we  have,  by  Napier's  rule, 

R  sin.  CD = sin.  AC  sin,  A. 
Also,  in  the  triangle  BCD,  We  have 

R  sin.  CD=sin.  BC  sin.  B. 
Hence         sin  AC  sin.  A=sin.  BC  sin.  B, 
or  sin.  BC  :  sin.  AC  : :  sin.  A  :  sin.  B. 

(216.)  Cor.  1.  In  any  spherical  triangle,  the  cosines  of  tkt 
sides  are  proportional  to  the  cosines  of^the  segments  of  tht. 
base,  made  by  a  perpendicular  from  the  opposite  angle. 


164  TRIGONOMETRY. 

Fo/,  by  Theorem  I.,  Cor.  2, 

cos.  CD  :  R  :  :  cos.  AC  :  cos.  AD. 
Also,  cos.  CD  :  R  :  :  cos.  BC  :  cos.  BD. 

Hence          cos.  AC  :  cos.  BC  : :  cos.  AD  :  cos.  BD. 
Cor.  2.   The  cosines  of  the  angles  at  the  base  are  proper* 
tional  to  the  sines  of  the  segments  of  the  vertical  angle. 
For,  "by  Theorem  I.,  Cor.  3, 

cos.  CD  :  R  :  :  cos.  A  :  sin.  ACD. 
Also,  cos.  CD  :  R  :  :  cos.  B  :  sin.  BCD. 

Hence         cos.  A :  cos.  B  :  :  sin.  ACD  :  sin.  BCD. 

<     Cor.  3.   The  sines  of  the  segments  of  the  base  are  recipr> 
rally  proportional  to  the  tangents  of  the  angles  at  the  bas? 

For,  by  Theorem  II., 

sin.  AD  :  R  :  :  tan.  CD  :  tan.  A. 

Also,  sin.  BD  :  R  :  :  tan.  CD  :  tan.  B. 

Hence  sin.  AD  :  sin.  BD  :  :  tan.  B  :  tan.  A. 

Cor.  4.  The  cotangents  of  the  two  sides  are  proportions 
>.o  the  cosines  of  the  segments  of  the  vertical  angle. 

For,  by  Theorem  II.,  Cor.  2, 

cos.  ACD  :  cot.  AC  :  :  tan.  CD  :  R. 

Also,         cos.  BCD  :  cot.  BC  :  :  tan.  CD  :  R. 

Hence       cos.  ACD  :  cos.  BCD  :  :  cot.  AC  :  cot.  BC. 

THEOREM  IV. 

(217.)  If  from  an  angle  of  a  spherical  triangle  a  perpen- 
(Uvular  be  drawn  to  the  base,  then  the  tangent  of  half  the  sum 
of  the  segments  of  the  base  is  to  the  tangent  of  half  the  sum 
of  the  sides,  as  the  tangent  of  half  the  difference  of  the  sides 
is  to  the  tangent  of  half  the  difference  of  the  segments  of  th* 
base. 

Let  ABC  be  any  spherical  trian- 
gle, and  let  CD  be  drawn  from  C 
perpendicular  to  the  base  AB  ;  then 
tan.  i(BD+AD) :  tan.  ^(BC+AC) : : 
tan.  J(BC-AC)  :  tan.  i(BD-AD). 

Let  BC=o,  AC =6,  BD=m,  and 
ATh=».  Then,  by  Theorem  III.,  Cor.  1, 

cos.  a  :  cos.  b  :  :  cos.  m  :  cos.  ». 


SPHERICAL   TRIGONOMETRY  163 

Whoncs,  G-eom.,  Prop.  7,  Cor.,  B.  IT., 
Gets.  6+C03.  a  :  cos.  b— cos.  a  :  :  cos.  w-fcos.  m  :  cos,  n— cos.  m 

But  by  Trig. ,  Art.  76, 

cos.  &+GOS.  a  :  cos.  b— cos.  a  :  :  cot.  \(a-\-b)  :  tan.  ^(a— &). 

Also,  by  the  same  Art., 
cos.  «+cos.  m  :  cos.  n— cos.  m  :  :  cot.  J(m+w)  :  tan.  J(i»— w). 

Therefore 

cot.  |(#+&)  :  cot.  J(w-r-ft)  :  :  tan.  ^(a— b)  :  tan.  $(m—n). 

But,  since  tangents  are  reciprocally  as  their  cotangents,  Art. 
88,  we  have 

cot.  ^(a+b)  :  cot.  \(m-\-ri)  :  :  tan.  \(m+n)  :  tan.  ^(a+b) 

Hence 

tan.  £(w+w)  :  tan.  \(a+b)  :  :  tan.  \(a  —  b)  :  tan.  %(m— n). 

(218.)  In  the  solution  of  oblique-angled  spherical  triangle*, 
six  cases  may  occur,  viz, : 

1.  (riven  two  sides  and  an  angle  opposite  one  of  them. 

2.  Griven  two  angles  and  a  side  opposite  one  of  them. 

3.  Griven  two  sides  and  the  included  angle. 

4.  Griven  two  angles  and  the  included  side. 

5.  Given  the  three  sides, 
^5.  Griven  the  three  angles. 

CASE  I. 

(219.)  Given  two  sides  and  an  angle  opposite  one  of  them^ 
io  find  the  remaining1  parts. 

In  the  triangle  ABC,  let  there  be  given  the  two  sides  AC 
and  BC,  and  the  angle  A  opposite  one 
of  them.    The  angle  B  may  be  found 
by  Theorem  III. 
sin.  BC  :  sin.  AC  : :  sin.  A  :  sin.  B. 

From  the  angle  C  let  fall  the  per- 
pendicular  CD  upon  the  side  AB. 
The  triangle  ABC  is  divided  into  two  right-angled  triangles, 
in  each  of  which  there  is  given  the  hypothenuse  and  the  angla 
at  the  base.  The  remaining  parts  may  then  be  found  by  Na- 
rule. 

Ex.  1.  In  the  oblique-angled  sphericil  triangle  ABC,  the 


I  b'ti  TRIGONOMETRY. 

*ide  AC=70°  10  30",  BC=80°  5'  4",  and  the  angle  A^53f 
1.5'  7".     Required  the  other  parts. 

sin  BC  :  sin.  AC  :  :  sin.  A  :  sin.  B=31°  34'  38" 
Then,  in  the  triangle  ACD, 

R  cos.  AC = cot.  A  cot.  ACD. 
Whence  ACD =77°  27'  47". 

Also,  in  the  triangle  BCD, 

R  cos.  BC=cot.  B  cot.  BCD. 
Whence  BCD=  83°  57'  29". 

Therefore  ACB=161°  25'  16". 

To  find  the  side  AB. 

sin.  A  :  sin.  ACB  : :  sin.  BC  :  sin.  AB=145°  5'  0". 

When  we  have  given  two  sides  and  an  opposite  angle,  there 
are,  in  general,  two  solutions,  each  of  which  will  satisfy  the 
conditions  of  the  problem.  If  the  side  AC,  the  angle  A,  and 
the  side  opposite  this  angle  are  given, 
then,  with  the  latter  for  radius,  de- 
scribe an  arc  cutting  the  arc  AB  in 
the  points  B  and  B'.  The  arcs  CB, 
CB'  will  be  equal,  and  each  of  the  tri- 
angles ACB,  ACB'  will  satisfy  the  B' 
conditions  of  the  problem.  There  is  the  same  ambiguity  in 
the  numerical  computation.  The  angle  B  is  found  by  means 
of  its  sine.  But  this  may  be  the  sine  either  of  ABC,  or  of  its 
supplement  AB'C  (Art.  27).  In  the  preceding  example,  the 
first  proportion  leaves  it  ambiguous  whether  the  angle  B  is 
31°  84'  38',  or  its  supplement  148°  25'  22".  In  order  to  avoid 
false  solutions,  we  should  remember  that  the  greater  side  of 
a  spherical  triangle  must  lie  opposite  the  greater  angle,  and 
conversely  (Geom.,  Prop.  17,  B.  IX.).  Thus,  since  in  the  pre- 
ceding example  the  side  AC  is  less  than  BC,  the  angle  B  must 
be  less  than  A,  and,  therefore,  can  not  be  obtuse. 

If  the  quantity  sought  is  determined  by  means  of  its  cosine, 
tangent,  or  cotangent,  the  algebraic  sign  of  the  result  will 
show  whether  this  quantity  is  less  or  greater  than  90°  ;  for  the 
cosines,  tangents,  and  cotangents  are  positive  in  the  first  quad- 
rant,  and  negative  in  the  second.  Hence  the  algebraic  sign 


SPHERICAL    TRIGONCMETR\.  16* 

of  each  term  of  a  proportion  should  be  preserved  whenever  ona 
of  them  is  negative. 

Ex.  2.  In  the  spherical  triangle  ABC,  the  side  0=124°  53', 
6=31°  19',  and  the  angle  A=16°  26'.  Required  the  remain- 
ing  parts. 

(  B=  10°  19'  34" 

Ans.  }  C=171°  48'  22" 

(  c  =-155°  35'  22' 

CASE  II. 

(220.)  Given  two  angles  and  a  side  opposite  one  of  them^ 
to  find  the  remaining  parts. 

In  the  triangle  ABC  let  there  be  given  two  angles,  as  A  and 
B,  and  the  side  AC  opposite  to  one 
of  them.  The  side  BC  may  be 
found  by  Theorem  III. 
sin.  B  :  sin.  A  : :  sin.  AC  :  sin.  BC. 
From  the  unknown  angle  C  draw 
CD  perpendicular  to  AB  ;  then  will 
the  triangle  ABC  be  divided  into  two  right-angled  triangles,  in 
each  of  which  there  is  given  the  hypothenuse  and  the  angle  at 
the  base.  Whence  we  may  proceed  by  Napier's  rule,  as  in 
Case  I. 

Ex.  1.  In  the  oblique-angled  spherical  triangle  A.BC,  there 
are  given  the  angle  A=52°  20',  B=63°  40',  and  the  sidu 
/;=S3°  25'.  Required  the  remaining  parts. 

sin.  B  :  sin.  A  :  :  sin.  AC  :  sin.  BC=61°  19'  53". 
Then,  in  the  triangle  ACD, 

cot.  AC  :  R  :  :  cos.  A  :  tan.  AD =79°  18'  17". 
Also,  in  the  triangle  BCD, 

cot.  BC  :  R  :  :  cos.  B  :  tan.  BD=39°  3'  8". 
Hence  AB=118°  21'  25". 

To  find  the  angle  ACB. 
sin.  BC  :  sin.  AB  :  :  sin.  A  :  sin.  ACB=127°  26'  47". 

When  we  have  given  twc  angles  and  an  opposite  side,  there 
are,  in  general,  two  solutions,  each  of  which  will  satisfy  the 
conditions  of  the  problem.  If  the  angle  A,  the  side  AC,  and 


168  TRIGONOMETRY. 

the  fcngle  opposite  this  side  are  given,  then  through  the  point 
C  there  may  generally  be 
drawn  two  arcs  of  great  cir- 
cles CB,  CB',  making  the 
same  angle  with  AB,  and 
each  of  the  triangles  ABC, 
AB'C  will  satisfy  the  condi- 
tions of  the  problem.  There  is  the  same  ambiguity  in  the 
numerical  computation,  since  the  side  BC  is  found  by  means 
of  its  sine  (Art.  27).  In  the  preceding  example,  however, 
there  is  no  ambiguity,  because  the  angle  A  is  less  than  B, 
and,  therefore,  the  side  a  must  be  less  than  Z>,  that  is,  less  than 
a  quadrant. 

Ex.  2.  In  the  oblique-angled  spherical  triangle  ABC,  the 
angle  A  is  128°  45',  the  angle  C=30°35/,  and  BC  =  6S°50'. 
Required  the  remaining  parts. 

It  will  be  observed  that  in  this  case  the  perpendicular  BD5 
drawn  from  the  angle  B,  falls  without  the  triangle  ABC,  and 
therefore  the  side  AC  is  the  difference  between  the  segment* 
CD  and  AD.  •(  AB=37°  28'  20  ' 

Am.  \  AC -40°    9'    4". 
(  B    =32°  37'  58". 

CASE  III. 

(221.)  Given  two  sides  and  the  included  angle,  to  find  tk 
remaining  parts. 

In  the  triangle  ABC  let  there  be  given  two  sides,  as  AB 
AC,  and  the  included  angle  A.     Let 
fall  the  perpendicular  CD  on  the  side 
AB  ;  then,  by  Napier's  rule, 

R  cos.  A=tan.  AD  cot.  AC. 

Having  found  the  segment  AD,  the 
segment  BD  becomes  known ;  then, 
by  Theorem  III.,  Cor.  3, 

sin.  BD  :  sin.  AD  :  :  tan.  A  :  tan.  B. 

The  remaining  parts  may  now  be  found  by  Theorem  III. 

Ex.  1.  In  the  spherical  triangle  ABC,  the  side  AB  =  73°  20  , 
AC =41°  45',  and  the  angle  A^30°  30'.  Required  the  remain- 
ing par's 


SPHERICAL    TRIGONOMETRY.  16U 

cot.  AC  :  cos.  A  :  :  R  :  tan.  AD=37°  33'  41". 
Hence     .  '  BD=35°  46'  19". 

sin.  BD  :  sin.  AD  •  :  tan.  A.  :  tan.  B=31°  33'  43". 
Also,  by  Theorem  III.,  Cor.  1, 

cos.  AD  :  cos.  BD  :  :  cos.  AC  :  cos.  BC-40'  J3X  0/x. 
Then,  by  Theorem  III., 

sin.  BC  :  sin.  AB  :  :  sin.  A  :  sin.  ACB=131°  S'  47'  . 
Ex.  2.  In  the  spherical  triangle  ABC,  the  side  AB=78°  15', 
A.C=56°  20',  and  the  angle  A=l?0°.     Required  the  othti 
parts. 

(  B   -=48°  57'  29". 
Ans.  }  C   =62°  31'  40". 
7'  45' 


CASE  IV. 

(222.)  Given  two  angles  and  the  included  side,  to  find  ttit 
remaining  parts. 

In  the  triangle  ABC  let  there  be  given  two  angles,  as  A  and 
ACB,  and  the  side  AC  included  be- 
tween them.     From  C  let  fall  the  per- 
pendicular CD  on  the  side  AB.     Then, 
by  Napier's  rule, 

R  cos.  AC=cot.  A  cot.  ACD. 
Havirigr  found  the  angle  ACD,  the 
angle  BCD  becomes  known  ;  then,  by 
Theorem  III.,  Cor.  4, 

cos.  ACD  :  cos.  BCD  :  :  cot.  AC  :  cot.  BC. 
The  remaining  parts  may  now  be  found  by  Theorem  111. 
Ex.  1.  In  the  spherical  triangle  ABC,  the  angle  A=32°  10', 
the  angle  ACB  =133°  20',  and  the  side  AC  =39°  15'.     R* 
quired  the  other  parts. 

cot.  A  :  cos.  AC  :  :  R  :  cot.  ACD=64°  1'  57" 
Hence  BCD=69°  18'  3". 

Then 

cos.  ACD  :  cos.  BCD  :  :  cot.  AC  :  cot.  BC=45D  20'  43'  , 
Also,  by  Theorem  III.,  Cor.  2, 

sin.  ACD  :  sin.  BCD  :  :  cos.  A  :  cos.  B=28°  15'  47". 
Then,  by  Theorem  III., 

sin.  B  :  sin  ACB  :  :  sin,  AC  :  sin.  AB-^760  23'  Ti". 


170  T  R  I  G  0  N  O  to  E  I'  R  V 

Ex.  2.  In  the  spherical  triangle  ABC,  the  angle  A=125°  20 \ 
the  angle  C=48'J  30',  and  the  side  AC=83°  13'.  Recmired 
fhe  remaining  parts. 

(  AB=  56°  39'    9". 

Ans.  }BC  =  114°30'24".: 

(  B    =62°  54'  SS  . 

CASE  V. 

(223.)  Given  the  three  sides  of  a  spherical  triangle,  to  find 
the  angles. 

In  the  triangle  ABC  let  there  be  given  the  three  side*. 
From  one  of  the  angles,  as  C,  draw  CD 
perpendicular  to  AB.     Then,  by  The- 
orem IV.,  tan.  JAB  :  tan.  4(AC+BC)  : 
tan.  i(AC-BC)  :  tan.  l(AD-BD). 

Hence  AD  and  BD  become  known ; 
then,  by  Napier's  rule, 

R  cos.  A=tan.  AD  cot.  AC. 
The  other  angles  may  now  be  easily  found. 
It  is  generally  most  convenient  to  let  fall  the  perpendicular 
upon  the  longest  side  of  the  triangle. 

Ex.  1.  In  the  spherical  triangle  ABC,  the  side  AB=1123 
25 ,  AC =60°  20',  and  BC=81°  10'.     Required  the  angles, 
tin.  56°  12J-'  :  tan.  70°  45'  : :  tan.  10°  25'  :  tan.  19°  24'  26''. 
Hence     AD=36°  48'  4",  and  BD=75°  36'  56". 
Then    R  :  tan.  AD  :  :  cot.  AC  :  cos.  A=64°  46'  36". 
Also,     R  :  tan.  BD  :  :  cot.  BC  :  cos.  B=52°  42'  12". 
Then  sin.  AC  :  sin.  AB  :  :  sin.  B  :  sin.  ACB=122°  11'  6". 
Ex.  2.  In  the  spherical  triangle  ABC,  the  side  AB=40°  35 
AC -39°  10',  and  BC=71°  15 .     Required  the  angles. 

(  A=130°  35'  55". 

Ans.  )-B=  30°  25' 34' 

(  C-  31°  26'  32". 

CASE  VI. 

(224.)  Given  the  three  angles  of  a  spherical  triangle,  to 
find  the  side*. 

If  A,  B,  C  are  the  angles  of  the  given  triangle,  and  a,  b.  c 
its  sides,  then  180° -A,  180° -B,  and  180° -C  are  the  sides 


SPHERICAL   TRIGONOMETRY.  17i 

ci  its  polar  triangle,  whose  angles  may  be  found  by  Case  V 
Then  the  supplements  of  those  angles  will  be  the  sides  a,  b,  ( 
of  the  proposed  triangle. 

Ex.  1.  In  the  spherical  triangle  ABC,  the  angle  A=125C 
34',  B=98°  44',  and  0=61°'  53'.     Required  the  sides. 
The  sides  of  the  polar  triangle  are 

54°  26',  81°  16',  and  118°  7'. 
f'rom  which,  by  Case  V.,  the  angles  are  found  to  be 

134°  6'  21",  41°  28'  17",  and  53°  34'  47". 
Hence  the  sides  of  the  proposed  triangle  are 
AB=45°  53'  39",  BC=138°  31'  43",  and  AC-1260  25'  13" 

Ex.  2.  In  the  spherical  triangle  ABC,  the  angle  A=1093 
55',  B=116°  38';  and  C=120°  43'.     Required  the  sides. 

(  a=  98°  21'  20'  . 
Ans.     6=109°  50'  10". 


TRIGONOMETRICAL  FORMULAS. 

(225.)  Let  ABC  be  any  spherical 
triangle,  and  from  the  angle  B  draw 
the  arc  BD  perpendicular  to  the  base 
AC.  Represent  the  sides  of  the  trian- 
gle by  a,  b,  c,  and  the  segment  AD  by 
x ;  then  will  CD  be  equal  to  b— x. 

By  Theorem  III.,  Cor.,  1, 
cos.  c  :  cos.  a  :  :  cos.  x  :  cos.  (b— x) 

cos.  b  cos.  re + sin.  b  sin.  x 
:  :  cos.  x  :  ^5 

(Trig.,  Art.  72),  formula  (4). 

Whence 

R  cos.  a  cos.  re— cos.  b  cos.  c  cos.  rc+sin.  b  cos.  c  sin.  x t 
or,  dividing  each  term  by  cos.  x,  and  substituting  the  value  «rf 


(Art.  28\  we  obtain 

V 


COS.  X 

R8  cos.  #=R  cos.  b  cos.  c-f-  sin.  b  cos.  c  tan, 


But  by  Theorem  II.,  Cor.  2,  we  have 
I  cos.  A_^cos.  A  sh 
cot.  c  cos.  c 


R  cos.  A    cos.  A  sin.  c 
tan.  x=-        —  --     (Art,  28). 

f\r\4r      /*  r*f\<3      /•  *  * 


172  TRIGONOMETRY. 

Hence  R2  cos.  a=R  cos.  b  cos.  c+sin.  b  sin.  c  cos.  A,  (1; 
from  which  all  the  formulas  necessary  for  the  solution  of  spheri- 
cal triangles  may  be  deduced. 

Tn  a  similar  manner  we  obtain 

R2  cos.  &=R  cos.  a  cos.  c+sin.  a  sin.  c  cos.  B,         (2) 
Ra  cos.  c=R  cos.  a  cos.  #+sin.  a  sin.  b  cos.  C.         (3) 

These  equations  express  the  following  Theorem : 

The  square  of  radius  multiplied  by  the  cosine  of  either  side 
of  a  spherical  triangle,  is  equal  to  radius  into  the  product  of 
thz  cosines  of  the  two  other  sides,  plus  the  product  of  the  sines 
of  those  sides  into  the  cosine  of  their  included  angh 

(226.)   From  equation  (1)  we  obtain,  by  transposition, 

R2  cos.  a— R  cos.  b  cos.  c 

cos.  A= : — - — : , 

sin.  b  sm.  c 

a  formula  which  furnishes  an  angle  of  a  triangle  when  the 
three  sides  are  known. 

If  we  add  R  to  each  member  of  this  equation,  we  shall  have 
_  R2  cos.  &+R  sin.  b  sin.  c— R  cos.  b  cos.  c 

R  +  COS.  A  =  —  — : ; : 

sin.  b  sin.  c 

O      f>/^Q  _1_  A 

But,  by  Art.  74,  R+cos.  A= ^~~' 

And,  by  Art.  72,  formula  (2),  by  transposition, 

R  sin.  b  sin.  c— R  cos.  b  cos.  c=—  R2  cos.  (£+c). 
Hence,  by  substitution,  we  obtain 

2  cos.  2JA_R2(cos.  a— cos.  (b+c)) 
R  sin.  b  sin.  c 

_2R  sin,  ^(a+b+c)  sin,  ^(b+c-a) 

sin.  b  sin.  c 
by  Art.  75,  formula  (4). 

If,  then,  we  put  s=^(a+b+c),  that  is,  half  the  sum  of  tha 
sides,  we  shall  find 

_  .  /sin.  s  sin.  (s  —  a) 

cos.  AA=RV = — i — • '•  (4) 

sm.  b  sin.  c 

By  subtracting  cos.  A  from  R  instead  of  adding,  we  shal 
obtain,  in  a  similar  manner, 


sn.    = 

sin.  b 


SPHERICAL    TRIGONOMETRY.  17£ 

Either  formula  (4)  or  (5)  may  be  employed  to  compute  thr 
angles  of  a  spherical  triangle  when  the  three  sides  are  knovfn, 
and  this  method  may  be  preferred  to  that  of  Art.  223. 

Ex.  1.  In  a  spherical  triangle  there  are  given  a =63°  50', 
6=80°  19',  and  c=120°  47'.     Required  the  three  angles. 
Here  half  the  sum  of  the  sides  is  132°  28' =s. 
Also,  s-a=68°  38'. 

Using  formula  (4),  we  have 

log.  sine  s,  132°  28'     .     .     9.867862 

log.  sine  (s- a),  68°  38'  .  .  9.969075 
-log.  sine  6,  80°  19'  comp.  0.006232 
-log.  sine  c,  120°  47'  comp.  0.065952 

Sum     19.909121 

log.  cos.  JA,          25°  45'  19"        9.954560. 
Elence  the  angle  A=51°  30'  38". 

The  remaining  angles  may  be  found  by  Theorem  TIL,  or  by 
formulas  similar  to  formula  (4). 


cos.  %n—. 

sin.  a  sm.  c 

.,        _     /si 
cos.  1C— . 


sin.  a  sin.  b 
We  thus  find  the  angle  B=  59°  16'  46", 
and  0  =  131°  28'  36". 

Ex.  2.  In  a  spherical  triangle  there  are  given  #=115°  20', 
/>=57°  30',  and  c=S2°  28'.  Required  the  three  angles. 

(  A=126°  35'    2". 

Arcs.  }  B=  48°  31'  42". 

(  C=  61°  43'  58". 

(227.)  By  means  of  the  polar  triangle,  we  may  convert  the 
preceding  formulae  for  angles  into  formulae  for  the  sides  of  a 
triangle,  since  the  angles  of  every  triangle  are  the  supplements 
of  the  sides  of  its  polar  triangle.  Let,  then,  a',  b',  c',  A',  B', 
C'  represent  the  sides  and  angles  of  the  polar  triangle,  and  we 
shall  have 

A=180°  —  #',  B  =  1SO°  —  b',  C  = 
a=180°-A',  &=180°-B',  c= 
Therefore        sin.  JA=sin.  (90°— \a')  =cos. 
cos.  |A  =cos.  (90°-|a')  =sin. 


174  TRIGONOMETRY. 

sin.  b=s'm.  (180°-B'}=sin.  B', 
sin.  c=sin.  (180°-C;)=sin.  C'. 

Also,  if  we  put  S'=half  the  sum  of  the  angles  t»f  the  poia* 
triangle,  we,  shall  have 


or  s=270°-S', 

whence  sin.  s=—  cos.  S', 

sin.  (s-a)=sin.  [90°-(S'-A')]=cos.  (S'-A'^ 

sin.  (s-b)=cos.  (S'-B'), 

sin.  (s-c)=cos.  (S'-C'). 

By  substituting  these  values  in  formula  (5),  Art.  226,  and 
omitting  all  the  accents,  since  the  equations  are  applicable  to 
any  triangle,  we  obtain 


.       _       cos.  (S-B)cos.(S-C) 
cos.  Ja=R\    -  ^  —  ^  --  ^  --  ';  (6) 

sin.  B  sm.  C 


and  formula  (4)  becomes 


sS  cos.  (S-A) 

sin.  ia=R\-  p    .  vn  -  -;,  (7) 

sin.  B  sin.  C 

which  formulae  enable  us  to  compute  the  sides  of  a  triangle 
when  the  three  angles  are  known  ;  and  this  method  may  bo 
preferred  to  that  of  Art.  224. 

In  a  similar  manner,  by  means  of  the  polar  triangle,  we 
rlerive  from  formula  (1),  Art.  225,  the  equation 

Ra  cos.  A=cos.  a  sin.  B  sin.  C  —  R  cos.  13  cos.  C  ;     (8) 

that  is,  the  square  of  radius  multiplied  by  the  cosine  of  either 
angle  of  a  spherical  triangle,  is  equal  to  the  product  of  the 
sines  of  the  two  other  angles  into  the  cosine  of  their  included 
side,  minus  radius  into  the  product  of  their  cosines. 

Ex.  1.  In  a  spherical  triangle  ABC,  there  are  given  A— 
130°  30',  B=30°  50',  and  0=32°  5'.  Required  the  Hire* 
sides. 

Here  half  the  sum  of  the  angles  is  96°  42  30''=^S. 

Also,  S-A=-33°47'30", 

S-B=     65°  52'  30", 
S-C=     6/°37'30' 

Using  formula  (6),  we  have 


SPHERICAL   TRIGONOMETRY.  .75 

log,  cos.  (S-B),  65°  52'  30"     .  9.611435 

log.  cos.  (S-C),  64°  37'  30"     .  9.631992 

-log.  sin.  B,       30°  50'  comp.  0.290270 

-log.  sin.  C,       32°    5'  comp.  0.274781 

Sum  19.808478 

log.  cos.  \a,        36°  40'  1"  9.904239. 
Hence  the  side  a=73°  20'  2". 

The  remaining  sides  may  be  found  by  Theorem  TIL,  or  by 
formulas  similar  to  formula  (6). 

4  /cos.  (S-A)cos.  (S-C) 

COS.  \b  —  ]^  V  : : ~ , 

sin.  A  sin.  C 


.       _     /cos.  (S-A)  cos.  (S-B) 

cos.  ^C=RV — ^ — T—' — 5 — • 

sin.  A  sin.  i3 

We  thus  find  the  side  6=40°  13°  12", 
and  c=42°    0'  12". 

Ex.  2.  In  the  spherical  triangle  ABC,  the  angle  A=129: 
30',  B=54°  35',  and  C  =  63°  5'.  Required  the  three  sides. 

(  0=120°  57'    5". 

Ans.  ]b=  64°  55' 37" 

(c  =  82°  19'    0". 

(228.)  Formula  (1),  Art.  225,  will  also  furnish  a  new  test 
for  removing  the  ambiguity  of  the  solution  in  Case  I.  of  oblique 
angled  triangles.  For  we  have 

o  O  .  - 

R2  cos.  a— R  cos.  b  cos.  c 

COS.   A== : ; : . 

sm.  b  sin.  c 
Now  if  cos.  a  is  greater  than  cos.  b,  we  shall  have 

R2  cos.  «>R  cos.  b  cos.  c, 

or  the  sign  of  the  second  member  of  the  equation  will  be  the 
same  as  that  of  cos.  a,  since  the  denominator  is  necessarily 
positive,  and  cos.  c  is  less  than  radius.  Hence  cos.  A  and  cos. 
a  will  have  the  same  sign ;  or  A  and  a  will  be  of  the  sama 
species  when  cos  #>cos.  b,  or  sin.  a<sin.  b ;  that  is, 

If  the  sine  of  the  side  opposite  to  the  required  angle  is  less 
than  the  sine  of  the  other  given  side,  there  will  be  but  one 
triangle. 

But  if  cos.  a  is  less  thai*,  cos.  &,  then  such  a  value  may  be 
pjiven  to  c  as  to  render 


17^  TRIGONOMETRY. 

R2  cos.  a<R  cos.  b  cos.  c, 

or  the  sign  of  the  second  member  of  the  equation  will  depend 
upon  the  value  of  cos.  c  ;  that  is,  c  may  be  taken  so  as  to  ren- 
der cos.  A  either  positive  or  negative.  Hence 

If  the  sine  of  the  side  opposite,  to  the  required  angle  u 
greater  than  the  sine  of  the  other  given  side,  there  will  be 
two  triangles  which  fulfill  the  given  conditions. 

(229.)  Formula  (8),  Art.  227,  will  furnish  a  t.  st  for  ro- 
moving  the  ambiguity  in  Case  II.  of  oblique-angled  triangles. 
For  we  have 

R2  cos.  A+R  cos.  B  cos.  C 

cos.  a= : — = — : — •= ; 

sin.  B  sin.  C 

trom  which  it  follows,  as  in  the  preceding  article,  that  if  cos.  A 
is  greater  than  cos.  B,  A  and  a  will  be  of  the  same  species.  But 
if  cos.  A  is  less  than  cos.  B,  then  such  values  may  be  given 
to  C  as  to  render  cos.  a  either  positive  or  negative.  Hence 

If  the  sine  of  the  angle  opposite  to  the  required  side  is  less 
than  the  sine  of  the  other  given  angle,  there  will  be  but  om 
triangle  ; 

But,  if  the  sine  of  the  angle  opposite  to  the  required  sick 
is  greater  than  the  sine  of  the  other  given  angle,  there  wih 
be  two  triangles  which  fulfill  the  given  conditions. 

SAILING  ON  AN  ARC  OF  A  GREAT  CIRCLE. 

(230.)  It  is  demonstrated  in  Geom.,  Prop.  3,  B.  IX.,  that  the 
shortest  path  from  one  point  to  another  on  the  surface  of  a 
sphere  is  the  arc  of  a  great  circle  which  joins  the  two  given 
points.  Hence,  if  it  is  desired  to  sail  from  one  port  to  another 
by  the  shortest  route,  it  is  necessary  to  follow  an  arc  of  a  great 
circle,  and  this  arc  generally  does  not  coincide  with  a  rhumb 
line. 

The  bearing  and  distance  from  one  place  to  another  on  the 
arc  of  a  great  circle  may  be  computed  from  the  latitudes  and 
longitudes  of  the  places  by  means  of  Spherical  Trigonometry 

Thus,  let  P  be  the  pole  of  the  earth,  EQ,  a  part  of  the  equa- 
tor, and  A  and  B  the  two  given  places  comprehended  between 
the  meridians  PE  and  PQ,.  Then  PA  is  the  complement  of 
the  latitude  of  A,  PB  is  1he  complement  of  the  latitude  of  "B 


SPHERICAL    T R I.G o N o M F, TRY. 


1/7 


and  the  angle  P  is  measured  by  the  arc  EQ,  which  ;s  tho 

difference   of  longitude   between   the   two 

places.     Hence,  in  the  triangle  ABP,  we 

have  given  two  sides  AP,  BP,  and  the  in- 

cluded angle  P,  from  which  we  may  com- 

pute the  side  AB,  and  the  angles  A  and  B, 

according  to  Case  III.  of  oblique-angled  tri- 

angles. 

Ex.  1.  Required  the  course  and  distance 
from  Nantucket  Shoals,  in  latitude  41°  4' 
N.,  longitude  69°  55'  W.,  to  Cape  Clear,  in 
latitude  51°  26'  N.,  longitude  9°  29'  W.,  on  the  arc  of  a 
circle. 

Here  we  have  given 

the  angle  P=69°  55'-  9°  29'  =60°  26'  ; 
the  side  PA=90°        -41°    4'=48°  56'  ; 
the  side  PB=90°        -51°  26'=3S°  34'. 
Then  cot.  PB  :  cos.  P  :  :  R  :  tan.  PD=21°  28'  35". 
Whence  AD=27°  27'  25". 

Also  sin.  AD  :  sin.  PD  :  :  tan.  P  :  tan.  A    =54°  27 
and        sin.  A     :  sin.  PB  :  :  sin.  P  :  sin.  AB=41°  47 
41°  47'  28"  is  equal  to  2507.47  nautical  miles. 
Hence  the  course  from  Nantucket 
Shoals  to  Cape  Clear  is  N.  54°  27' 
E.,  and  the  distance  is  2507.47  miles. 
According  to   Mercator's   sailing, 
the  course  on  a  rhumb-line,  found  on 
page  152,  is  N.  76°  E.,  and  the  dis- 
tance 2572.9  miles.     Hence  the  dis- 
tance on  an  arc  of  a  great  u'rclo  is  65.4  miles  less  than  on  a 
rhumb-line,  and  the  formei  course  is  2LJ  degrees  more  north- 
erly than  the  latter. 

"While  sailing  on  a  rhumb-line  the  course  of  a  ship  remains 
always  the  same,  but  while  sailing  on  an  arc  of  a  great  circle 
the  course  is  continually  changing.  The  preceding  course  is 
that  with  which  the  ship  starts  from  Nantucket,  and  a  new 
computation  of  the  oourse  should  be  made  every  day  or  two  ;  or 
.t  might  be  more  convenient  to  compute  beforehand  the  position 
of  the  points  in  which  the  great  circle  intersects  the  meridians 

M 


21 

28' 


178  TRIGONOMETRY 

for  every  five  degrees  of  longitude,  and  the  ship  might  t> 
steered  upon  a  direct  course  for  these  points  successively. 

Ex.  2.  Required  the  course  and  distance  from  Nantucket 
Shoals  to  G-ibraltar,  in  latitude  36°  6'  N.,  longitude  5°  2f  "V7., 
on  the  shortest  route. 

Ans.  The  course  is  N.  73°  29'  E 

Distance  2974.1  miles. 

Ex.  3.  Required  the  course  and  distance  from  Sandy  Hook, 
in  latitude  40°  28'  "N.,  longitude  74°  V  W.,  to  Madeira,  in 
latitude  32°  38'  K,  longitude  16°  55'  W.,  on  the  shortest  route. 

Ans.  The  course  is  N.  80°  53'  E. 

Distance  2744.1  miles. 

Ex.  4.  Required  the  course  and  distance  from  Sandy  Hoolf 
to  St.  Jago,  in  latitude  14°  54'  N.,  longitude  23°  30'  "W.,  on 
the  shortest  route. 

Ans.  The  course  is  S.  74°  46'  E 

Distance  3037.6  miles. 

Ex.  5.  Required  the  course  and  distance  from  Sandy  Hook 
to  the  Cape  of  Good  Hope,  in  latitude  34°  22'  S.,  longitude 
18°  30'  E.,  on  the  shortest  route. 

Ans.  The  course  is  S.  63°  48'  E 
Distance  6792  miles. 


EXAMPLES  FOR  PRACTICE. 

PLANE  TRIGONOMETRY. 

Prob.  1.  Given  the  three  sides  of  a  triangle,  627,  718.9,  and 
1140,  to  find  the  angles. 

Ans.  29°  44'  2",  34°  39'  26",  and  115°  36'  32". 
Prob.  2.  In  the  triangle  ABC,  the  angle  A  is  given  89°  45; 
43",  the  side  AB  654,  and  the  side  AC  460,  to  find  the  remain- 
ing parts. 

Ans.  BC  =  798;  the  angle  B  =  35°  12'  1",  and  the  angle 

C  =  55°2X  16". 

Prob.  3.  In  the  triangle  ABC,  the  angle  A  is  given  56°  12" 
45",  the  side  BC  2597.84,  and  the  side  AC  3084.33,  to  find  the 
remaining  parts. 

Ans.  B  =  80°  39'  40",  C  =  43°    T  35",  c  =  2136.8  ; 
or,  B  =  99    2020,    C=24    2655,    c  =  1293.8. 
Prob.  4.  In  the  triangle  ABC,  the  angle  A  is  given  44°  13' 
24",  the  angle  B  55°  59'  58".  and  the  side  AC  368,  to  find  the 
remaining  parts. 

Ans.  C  =  790  46'  38",  AB  =  436.844,  and  BC  =  309.595. 
Prob.  5.  In  a  right-angled  triangle,  if  the  surn  of  the  hy- 
pothenuse  and  base  be  3409  feet,  and  the  angle  at  the  base  53° 
12X  14",  what  is  the  perpendicular  ? 

Ans.  1707.2  feet. 

Prob.  6.  In  a  right-angled  triangle,  if  the  difference  of  the 
hypothenuse  and  base  be  169.9  yards,  and  the  angle  at  the  base 
42°  36'  12",  what  is  the  length  of  the  perpendicular  ? 

Ans.  435.732  yards. 

Prob.  7.  In  a  right-angled  triangle,  if  the  sum  of  the  base 
and  perpendicular  be  123.7  feet,  and  the  angle  at  the  base  58° 
19'  32",  what  is  the  length  of  the  hypothenuse  ? 

Ans.  89.889  feet. 

Prob.  8.  In  a  right-angled  triangle,  if  the  difference  of  the 
base  and  perpendicular  be  12  yards,  and  the  angle  at  the  base 
38°  V  8",  what  is  the  length  of  the  hypothenuse  ? 

Ans.  69.81  yards. 


180  TRIGONOMETRY. 

Prob.  9.  A  May-pole,  50  feet  11  inches  high,  at  a  certain 
time  will  cast  a  shadow  98  feet  6  inches  long;  what,  then,  is 
the  breadth  of  a  river  which  runs  within  20  feet  6  inches  of  the 
foot  of  a  steeple  300  feet  8  inches  high,  if  the  steeple  at  the 
same  time  throws  its  shadow  30  feet  9  inches  beyond  the  stream  ? 

Ans.  530  feet  5  inches. 

Prob.  10.  A  ladder  40  feet  long  may  be  so  placed  that  it  shall 
reach  a  window  33  feet  from  the  ground  on  one  side  of  the 
street,  and  by  turning  it  over,  without  moving  the  foot  out  of 
its  place,  it  will  do  the  same  by  a  window  21  feet  high  on  the 
other  side.  Required  the  breadth  of  the  street. 

Ans.  56.649  feet. 

Prob.  11.  A  May-pole,  whose  top  was  broken  off  by  a  blast 
of  wind,  struck  the  ground  at  the  distance  of  15  feet  from  the 
foot  of  the  pole  ;  what  was  the  height  of  the  whole  May-pole, 
supposing  the  length  of  the  broken  piece  to  be  39  feet  ? 

Ans.  75  feet. 

Prob.  12.  How  must  three  trees,  A,  B,  C,  be  planted,  so  that 
the  angle  at  A  may  be  double  the  angle  at  B,  the  angle  at  B 
double  the  angle  at  C,  and  a  line  of  400  yards  rnay  just  go 
round  them  ? 

Ans.  AB  =  79.225,  AC  =  142.758,  and  B  0  =  178.017  yards. 

Prob.  13.  The  town  B  is  half  way  between  the  towns  A  and 
C,  and  the  towns  B,  C,  and  D  are  equidistant  from  each  other. 
What  is  the  ratio  of  the  distance  AB  to  AD  ? 

Ans.  As  unity  to  <\/3. 

Prob.  14.  There  are  two  columns  left  standing  upright  in  the 
ruins  of  Persepolis  ;  the  one  is  66  feet  above  the  plain,  and  the 
other  48.  In  a  straight  line  between  them  stands  an  ancient 
statue,  the  head  of  which  is  100  feet  from  the  summit  of  the 
higher,  and  84  feet  from  the  top  of  the  lower  column,  the  base 
of  which  measures  just  74  feet  to  the  centre  of  the  figure's  base. 
Required  the  distance  between  the  tops  of  the  two  columns. 

Ans.  156.68  feet. 


Prob.  15.  Provo  that  tan.    45°-£)  = 


. 

Prob.  16.  One  angle  of  a  triangle  is  45°,  and  the  perpendic- 
ular from  this  angle  upon  the  opposite  base  divides  the  base  into 
two  parts,  which  are  in  the  ratio  of  2  to  3.  "What  are  the 


EXAMPLES   FOR   PRACTICE.  181 

parts  into  which  the  vertical  angle  is  divided  by  this  perpen- 
dicular? 

Ans.  18°  26'  6"  and  26°  33'  54". 
Prob.  17.  Prove  that  sin.  3b  —  3  sin.  &— 4  sin.3  b. 
Prob.  18.  One  side  of  a  triangle  is  25,  another  is  22,  and  the 
angle  contained  by  these  two  sides  is  one  half  of  the  angle  op- 
posite the  side  25.     What  is  the  value  of  the  included  angle? 

Ans.  39°  58'  51". 

Prob.  19.  One  side  of  a  triangle  is  25,  another  is  22,  and 
the  angle  contained  by  these  two  sides  is  one  half  of  the  angle 
opposite  the  side  22.  What  is  the  value  of  the  included  angle  ? 

Ans.  30°  46'  28".. 

Prob.  20.  Two  sides  of  a  triangle  are  in  the  ratio  of  11  to  9, 
and  the  opposite  angles  have  the  ratio  of  3  to  1.  What  are 
those  angles? 

Ans.  The  sine  of  the  smaller  of  the  two  angles  is  -J,  and  of 
the  greater  f-f ;  the  angles  are  41°  48X  37"  and 
125°  25'  51". 

Prob.  21.  One  side  of  a  triangle  is  15,  and  the  difference  of 
the  two  other  sides  is  6 ;  also,  the  angle  included  between  the 
first  side  and  the  greater  of  the  two  others  is  60°.  What  is  the 
length  of  the  side  opposite  to  this  angle  ? 

Ans.  57. 

Prob.  22.  One  side  of  a  triangle  is  15,  and  the  difference  of 
the  two  other  sides  is  6 ;  also,  the  angle  opposite  to  the  greater 
of  the  two  latter  sides  is  COC.  What  is  the  length  of  said  side  ? 

Ans.  13. 

Prob.  23.  One  side  of  a  triangle  is  15,  and  the  opposite  an- 
gle  is  60°  ;  also,  the  difference  of  the  two  other  sides  is  6 
What  are  the  lengths  of  those  sides  ? 

Ans.  11.0712  and  17.0712. 

Prob.  24.  The  perimeter  of  a  triangle  is  100  ;  the  perpendic- 
ular let  fall  from  one  of  the  angles  upon  the  opposite  base  is  30, 
and  the  angle  at  one  end  of  this  base  is  50°.  What  is  the 
length  of  the  base  ? 


182  TRIGONOMETRY. 

MENSURATION  OF  SURFACES  AND  SOLIDS. 

Prob.  1.  The  base  of  a  triangle  is  20  feet,  and  its  altitude 
18  feet.  It  is  required  to  draw  a  line  parallel  to  the  base  so  as 
to  cut  off  a  trapezoid  containing  80  square  feet.  What  is  the 
length  of  the  line  of  section,  and  its  distance  from  the  base  of 
the  triangle? 

-Ans.  Length  14.907  feet ;  distance  from  base  4.584  feet. 

Prob.  2.  The  base  of  a  triangle  is  20  feet,  one  angle  at  the 
base  is  63°  26',  and  the  other  angle  at  the  base  is  56°  19X.  It 
is  required  to  draw  a  line  parallel  to  the  base,  so  as  to  cut  off  a 
trapezoid  containing  109  square  feet.  What  is  the  length  of 
the  line  of  section,  and  its  distance  from  the  base  of  the  tri- 
angle ? 

Ans.  Length  12.070  feet;  distance  from  base  6.797  feet. 

Prob.  3.  In  a  perpendicular  section  of  a  ditch,  the  breadth  at 
the  top  is  26  feet,  the  slopes  of  the  sides  are  each  45°,  and  the 
area  140  square  feet.  Required  the  breadth  at  bottom  and  the 
depth  of  the  ditch. 

Ans.  Breadth  10.77  feet ;  depth  7.615  feet. 

Prob.  4.  The  altitude  of  a  trapezoid  is  23  feet ;  the  two  par- 
allel sides  are  76  and  36  feet ;  it  is  required  to  draw  a  line  par- 
allel to  the  parallel  sides,  so  as  to  cut  off  from  the  smaller  end 
of  the  trapezoid  a  part  containing  560  square  feet.  What 
is  the  length  of  the  line  of  section,  and  its  distance  from  the 
shorter  of  the  two  parallel  sides  ? 

Ans.  Length  56.954  feet;  distance  12.048  feet. 

Prob.  5.  From  the  greater  end  of  a  trapezoidal  field  whose 
parallel  ends  and  breadth  measure  12,  8,  and  10|  chains  re- 
spectively, it  is  required  to  cut  off  an  area  of  six  acres  by  a 
fence  parallel  to  the  parallel  sides  of  the  field.  What  is  the 
length  of  the  fence,  and  its  distance  from  the  greater  side. 

Ans.  Length  of  fence  9.914  chains;  distance  from  greater 
side  5.476  chains. 

Prob.  6.  There  are  three  circles  whose  radii  are  20,  28,  and 
29  inches  respectively.  Required  the  radius  of  a  fourth  circle, 
whose  area  is  equal  to  the  sum  of  the  areas  of  the  other  three. 

Ans.  45  inches, 

Prob.  7.  In  constructing  a  rail-road,  the  pathway  of  which 


EXAMPLES   FOR    PRACTICE.  183 

is  24  feet  broad,  it  is  necessary  to  make  a  cutting  40  feet  in 
depth;  what  must  be  the  breadth  of  the  cutting  at  top,  sup- 
posing the  slopes  of  the  sides  to  be  65°  ? 

Ans.  61.305  feet. 

Prob.  8.  The  sides  of  a  quadrilateral  field  are  690  yards, 
467  yards,  359  yards,  and  428  yards;  also,  the  angle  contained 
between  the  first  and  second  sides  is  57°  30',  and  the  angle  be- 
tween the  third  and  fourth  sides  96°  42'.  Required  the  area 
of  the  field. 

Ans.  212184  square  yards. 

Prob.  9.  There  are  two  regular  pentagons,  one  inscribed  in  a 
circle,  and  the  other  described  about  it ;  and  the  difference  of 
the  areas  of  the  pentagons  is  100  square  inches.  Required  the 
radius  of  the  circle. 

Ans.  8.926  inches. 

Prob.  10.  What  is  the  length  of  a  chord  cutting  off  one  third 
part  of  a  circle,  whose  diameter  is  289  feet. 

Ans.  278.67  feet. 

Prob.  11.  The  area  of  a  triangle  is  1012 ;  the  length  of  the 
side  a  is  to  that  of  b  as  4  to  3,  and  c  is  to  b  as  3  to  2.  Re- 
quired the  length  of  the  sides. 

Ans.  a  =  52.470,  6  =  39.353,  c  =  59.029. 
Prob.  12.  The  area  of  a  triangle  is  144,  the  base  is  24,  and 
one  of  the  angles  at  the  base  is  30°.     Required  the  other  sides 
of  the  triangle. 

Ans.  24  and  12.4233. 

V-  Prob.  13.  Seven  men  bought  a  grinding-stone  of  60  inches 
diameter,  each  paying  one  seventh  part  of  the  expense.     What 
part  of  the  diameter  must  each  grind  down  for  his  share  ? 
Ans.  The  1st, 4.4508  inches;  2d, 4.8400  inches;  3d, 5.3535 
inches ;  4th,  6.0765  inches ;  5th,  7.2079  inches ; 
6th,  9.3935  inches;  7th,  22.6778  inches. 
Prob.  14.  The  area  of  an  equilateral  triangle  is  17  square 
feet  and  83  square  inches.     What  is  the  length  of  each  side  ? 

Ans.  76.45  inches. 

Prob.  15.  The  parallel  sides  of  a  trapezoid  are  20  and  12  feet, 
and  the  other  sides  are  15  and  17  feet.  Required  the  area  of 
the  trapezoid. 

Ans.  240  square  feet. 


184  TRIGONOMETRY. 

Prob.  16.  How  many  square  yards  of  canvas  are  required  to 
make  a  conical  tent  which  is  20  feet  in  diameter  and  12  feet 
high? 

Am.  54.526  square  yards. 

Prob.  17.  The  circumference  of  an  hexagonal  pillar  is  7  feet, 
and  the  height  11  feet  2  inches.  Required  the  solid  contents  of 
the  pillar. 

Ans.  39.488  cubic  feet. 

Prob.  18.  The  base  of  the  great  pyramid  of  Egypt  is  a  square 
whose  side  measures  746  feet,  and  the  altitude  of  the  pyramid 
is  450  feet.  Required  the  volume  of  the  pyramid. 

Ans.  83,477,400  cubic  feet. 

Prob.  1 9.  A  side  of  the  base  of  a  frustum  of  a  square  pyra- 
mid is  25  inches,  a  side  of  the  top  is  9  inches,  and  the  height  is 
20  feet.  Required  the  volume  of  the  frustum. 

Ans.  43.102  cubic  feet. 

Prob.  20.  Three  persons,  having  bought  a  sugar-loaf,  would 
divide  it  equally  among  them  by  sections  parallel  to  the  base. 
It  is  required  to  find  the  altitude  of  each  person's  share,  suppos- 
ing the  loaf  to  be  a  cone  whose  height  is  20  inches. 

Ans.  13.8672,  3.6044,  and  2.5284  inches. 
Prob.  21.  If  a  cubical  foot  of  brass  were  to  be  drawn  into 
wire  of  one  thirtieth  of  an  inch  in  diameter,  it  is  required  to  de- 
termine the  length  of  the  said  wire,  allowing  no  loss  in  the  metal. 

Ans.  55003.94  yards ;  or  31  miles  443.94  yards. 
Prob.  22.  How  high  above  the  surface  of  the  earth  must  a 
person  be  raised  to  see  one  third  of  its  surface  ? 

Ans.  The  height  of  its  diameter. 

Prob.  23.  If  a  heavy  sphere,  whose  diameter  is  4  inches,  be 
let  fall  into  a  conical  glass  full  of  water,  whose  diameter  is  5, 
and  altitude  6  inches,  it  is  required  to  determine  how  much  wa- 
ter will  run  over. 

Ans.  26.272  cubic  inches. 

Prob.  24.  The  capacity  of  a  cylinder  is  a  cubic  feet,  and  its 
convex  surface  is  b  square  feet.  Required  the  dimensions  of 
the  cylinder. 

Ans.  Radius  of  base  =  — -,  and  altitude  = , 


Prob.  25.  A  triangular  pyramid,  the  sides  of  whose  base  are 


EXAMPLES   FOR    PRACTICE.  185 

13,  14,  and  15  inches  respectively,  and  whose  altitude  is  16 
inches,  is  cut,  at  the  distance  of  2  inches  from  the  vertex,  by  a 
plane  parallel  to  the  base.  Required  the  volume  of  the  frustum 
of  the  pyramid. 

Ans.  447.125  cubic  inches. 

Prob.  26.  The  altitude  of  a  cone  is  10  inches,  and  the  radius 
of  its  base  is  5  inches.  At  what  distance  from  the  base  must  a 
plane  pass  parallel  to  the  base,  so  as  to  cut  off  a  frustum  whose 
capacity  is  20  cubic  inches  ? 

Ans.  0.2614  inches. 

SURVEYING. 

Prob.  1.  The  angle  or  elevation  of  a  spire  I  found  to  be  39° 
27',  and  going  directly  from  it  225  feet  on  a  horizontal  plane,  I 
found  the  angle  to  be  only  24°  38'.     What  is  the  height  of  the 
spire,  and  the  distance  from  its  base  to  the  second  station  ? 
Ans.  Height  233.02  feet,  distance  508.18  feet. 

Prob.  2.  Wishing  to  know  the  distance  of  an  inaccessible  ob- 
ject, I  measured  a  horizontal  base-line  1328  feet,  and  found  the 
angles  at  the  ends  of  this  line  were  84°  23'  and  43°  19'.    What 
was  the  distance  of  the  object  from  each  end  of  the  base-line  ? 
Ans.  1151.44  feet,  and  1670.35  feet. 

Prob.  3.  Wishing  to  know  the  distance  between  two  inacces- 
sible objects,  C  and  D,  I  measured  a  base-line,  AB,  3784  feet, 
and  found  the  angle  BAD  =  47°  32',  the  angle  DAC  =  39°  53', 
the  angle  ABC =46°  34X,  and  the  angle  CBD^38°  V.  What 
is  the  distance  from  C  to  D  ? 

Ans.  3257.36  feet. 

Prob.  4.  Suppose  a  light-house  built  on  the  top  of  a  rock ;  the 
distance  between  the  place  of  observation  and  that  part  of  the 
rock  which  is  level  with  the  eye,  and  directly  under  the  build- 
ing, is  1860  feet ;  the  distance  from  the  top  of  the  rock  to  the 
place  of  observation  is  2538  feet,  and  from  the  top  of  the  build- 
ing 2550  feet.  Required  the  height  of  the  light-house. 

Ans.  17  feet  7  inches. 

Prob.  5.  At  85  feet  distance  from  the  bottom  of  a  tower, 
standing  on  a  horizontal  plane,  the  angle  of  its  elevation  was 
found  to  be  52°  30'.  Required  the  altitude  of  the  tower. 

Ans.  HOjJ  feet 


186  TRIGONOMETRY. 

Prob.  6.  At  a  certain  station,  the  angle  of  elevation  of  an  in- 
accessible tower  was  26°  30' ;  but,  measuring  225  feet  in  a  di- 
rect line  toward  it,  the  angle  was  then  found  to  be  51°  30'. 
Required  the  height  of  the  tower,  and  its  distance  from  the  last 
station.  Ans.  Height  186  feet,  distance  147  feet. 

Prob.  7.  To  find  the  distance  of  an  inaccessible  castle  gate,  I 
measured  a  line  of  73  yards,  and  at  each  end  of  it  took  the  an- 
gle of  position  of  the  object  and  the  other  end,  and  found  the  one 
to  be  90°,  and  the  other  61°  45'.  Required  the  distance  of  the 
castle  from  each  station. 

Ans.  135.8  yards,  and  154.2  yards. 

Prob.  8.  From  the  top  of  a  tower  143  feet  high,  by  the  sea- 
side, I  observed  that  the  angle  of  depression  of  a  boat  was  35°. 
What  was  its  distance  from  the  bottom  of  the  tower  ? 

Ans.  204.22  feet. 

Prob.  9.  I  wanted  to  know  the  distance  between  two  places, 
A  and  B,  but  could  not  meet  with  any  station  from  whence  I 
could  see  both  objects.  I  measured  a  line  CD  =  200  yards ;  from 
C  the  object  A  was  visible,  and  from  D  the  object  B  was  visi- 
ble, at  each  of  which  places  I  set  up  a  pole.  I  also  measured 
FC=200  yards,  and  DE=:200  yards,  and  at  F  and  E  set  up 
poles.  I  then  measured  the  angle  AFC  =  83°,  ACF  =  54°  31', 
ACD  =  53°  30',  BDC  =  156°  25',  BDE  =  54°  30',  and  BED=: 
88°  30'.  Required  the  distance  from  A  to  B. 

Ans.  345.5  yards. 

Prob.  10.  From  the  top  of  a  light-house,  the  angle  of  depres- 
sion of  a  ship  at  anchor  was  3°  38',  and  at  the  bottom  of  the 
light-house  the  angle  of  depression  was  2°  43'.  Required  the 
horizontal  distance  of  the  vessel,  and  the  height  of  the  promon- 
tory above  the  level  of  the  sea,  the  light-house  being  85  feet 
high.  Ans.  Distance  5296.4  feet,  height  251.3  feet. 

Prob.  11.  An  observer,  seeing  a  cloud  in  the  west,  measured 
its  angle  of  elevation,  and  found  it  to  be  64°.  A  second  observ- 
er, situated  half  a  mile  due  east  from  the  first  station,  and  on 
the  same  horizontal  plane,  found  its  angle  of  elevation  at  the 
same  moment  of  time  to  be  only  35°.  Required  the  perpendic- 
ular height  of  the  cloud,  and  its  distance  from  each  observer. 

Ans.  Perpendicular  height  935.75  yards,  distances  1041.1 
and  1631.4  yards. 


EXAMPLES   FOR   PRACTICE.  187 

Prob.  12.  An  observer,  seeing  a  balloon  in  the  north,,  meas- 
ured its  angle  of  elevation,  and  found  it  to  be  36°  52'.  A  second 
observer,  situated  one  mile  due  south  from  the  first  station,  and 
on  the  same  horizontal  plane,  found  its  angle. of  elevation  at  the 
same  instant  to  be  only  30°  58'.  Required  the  perpendicular 
height  of  the  balloon,  and  its  distance  from  each  observer. 
Ans.  Perpendicular  height  3.003  miles,  distances  5.006 

and  5.837  miles. 

Prob.  13.  From  a  window  near  the  bottom  of  a  house  which 
seemed  to  be  on  a  level  with  the  bottom  of  a  steeple,  I  found  the 
angle  of  elevation  of  the  top  of  the  steeple  to  be  40°  ;  then  from 
another  window,  21  feet  directly  above  the  former,  the  like  angle 
was  37°  30X.  "What  was  the  height  and  distance  of  the  stee- 
ple ?  Ans.  Height  245.51  feet,  distance  292.59  feet. 

Prob.  14.  "Wanting  to  know  my  distance  from  an  inaccessi- 
ble object,  P,  on  the  other  side  of  a  river,  and  having  no  instru- 
ment for  taking  angles,  but  a  chain  for  measuring  distances, 
from  eacli  of  two  stations,  A  and  B,  which  were  taken  at  300 
yards  asunder,  I  measured  in  a  direct  line  from  the  object  P  60 
yards,  viz.,  AC  and  BD  each  equal  to  60  yards  ;  also,  the  diag- 
onal AD  measured  330  yards,  and  the  diagonal  BC  336  yards. 
What  was  the  distance  of  the  object  P  from  each  station  A 
and  B  ?  Ans.  AP^ 321.76  yards,  BP  =  300.09  yards. 

Prob.  15.  Having  at  a  certain  (unknown)  distance  taken  the 
angle  of  elevation  of  a  steeple,  I  advanced  60  yards  nearer  on 
level  ground,  and  then  observed  the  angle  of  elevation  to  be  the 
complement  of  the  former.  Advancing  20  yards  still  nearer, 
the  angle  of  elevation  now  appeared  to  be  just  double  of  the 
first.  Required  the  altitude  of  the  steeple. 

Ans.  74.162  yards. 

Prob.  16.  In  a  garrison  there  are  three 
remarkable  objects,  A,  B,  C,  whose  dis- 
tances from  each  other  are  known  to  be, 
AB  213,  AC  424,  and  BC  262  yards.  I 
am  desirous  of  knowing  my  position  and- 
distance  at  a  station,  P,  from  which  I  ob- 
served the  angle  APB,  13°  30',  and  the 
angle  CPB,  29°  50'. 
Ans.  AP^  605.7122,  BP^  429,6814,  CP= 524.2365. 


188  TRIGONOMETRY. 

Prob.  17.  Supposing  the  object  B  to  be  on  the  opposite  side 
of  the  line  AC  (see  figure  to  Prob.  16),  and  that  the  distances  of 
the  objects  were,  AB  —  8  miles,  AC  =  12  miles,  and  BC  =  7-J  miles ; 
also,  the  angle  APB  =  19°,  and  the  angle  CPB  =  25°.  It  is  re- 
quired  to  find  the  distances  AP,  BP,  and  CP. 

Ans.  AP  =  9.4711  miles,  BP  =  16.3369  miles, 
CP  =  16.8485  miles. 

Prob.  18.  In  a  pentangular  field,  beginning  with  the  south 
side,  and  measuring  round  toward  the  east,  the  first  or  south 
side  was  27.35  chains,  the  second  31.15  chains,  the  third  23.70 
chains,  the  fourth  29.25  chains,  and  the  fifth  22.20  chains; 
also,  the  diagonal  from  the  first  angle  to  the  third  was  38.00 
chains,  and  that  from  the  third  to  the  fifth  was  40.10  chains. 
Required  the  area  of  the  field. 

Ans.  117  A.  2  R.  39  P. 

Prob.  19.  The  following  are  the  dimensions  of  a  five-sided 
field,  ABCDE:  the  side  AB  =  19.40  chains,  and  the  angle  B 
110°  30';  the  side  BC  =  15.55  chains,  and  the  angle  C  117° 
45' ;  the  side  CD  =  21.25  chains,  and  the  angle  D  91°  20' ;  and 
the  side  DE  =  27.41  chains.  Required  the  area  of  the  field. 

Ans.  66  A.  2  Jt.  24  P. 

Prob.  20.  From  a  station,  H,  near  the  middle  of  a  field, 
ABCDEF,  from  which  I  could  see  all  the  angles,  I  measured 
the  distances  to  the  several  corners,  and  measured  the  angles 
formed  at  H  by  those  distances,  as  follows : 

Distances.  Angles. 

AH,  43.15  chains ;  AHB,  60°  30'. 

BH,  29.82       "  BHC,  47    40 

CH,  35.61       "  CHD,  49    50 

DH,  50.10       "  DHE,57    10 

EH,  46.18       «  EHF,64    15 

FH,  36.06       «  FHA,  80    35 
Required  the  area  of  the  field. 

Ans.  412  A.  1  R.  17  P 

NAVIGATION. 

Prob.  1.  From  a  ship  at  sea  I  observed  a  point  of  land  to 
bear  east  by  south,  and,  after  sailing  northeast  12  miles,  I  ol> 


EXAMPLES   FOR   PRACTICE.  189 

served  again,  and  found  its  bearing  to  be  southeast  by  east. 
How  far  was  the  last  observation  made  from  the  point  of  land  ? 

Ans.  26.07  miles. 

Prob.  2.  If  a  ship  in  latitude  50°  N.,  sails  52  miles  in  the  di- 
rection southwest  by  south,  what  latitude  has  she  arrived  in, 
and  how  much  farther  to  the  west  ? 

Ans.  Latitude  49°  16/8  N. ;  west,  28.9  miles. 

Prob.  3.  Two  ships  sail  from  the  same  port ;  the  one  sails 
east-northeast  85  miles,  the  other  sails  east  by  south  till  the  first 
ship  bears  northwest  by  west.  What  is  the  distance  of  the  sec- 
ond ship  from  the  port,  and  also  from  the  first  ship  ? 

Ans.  From  the  port,  184.7  miles ;   from  the  first  ship, 
123.4  miles. 

Prob.  4.  Two  ports  lie  east  and  west  of  each  other ;  a  ship 
sails  from  each,  namely,  the  ship  from  the  west  port  sails  north- 
east 89  leagues,  and  the  other  sails  80  leagues,  when  she  meets 
the  former.  Required  the  latter  ship's  course,  and  the  distance 
between  the  two  ports. 

Ans.  Course,  N.  38°  8X  W. ;  distance,  112.3  leagues. 

Prob.  5.  Two  ships  sail  from  a  certain  port ;  the  one  sails 
south  by  east  45  leagues,  and  the  other  south-southwest  64 
leagues.  What  is  the  bearing  and  distance  of  the  first  ship  from 
the  second  ? 

Ans.  Bearing,  T\T.  65°  44'  E. ;  distance,  36.5  leagues. 

Prob.  6.  A  ship  sailing  northwest,  two  islands  appear  in 
sight,  of  which  the  one  bears  north,  and  the  other  west-north- 
west ;  but,  after  sailing  20  leagues,  the  former  bears  northeast, 
and  the  latter  west  by  south.  What  is  tlio  distance  asunder  of 
the  two  islands?  Ana.  32.38  leagues. 

Prob.  7.  To  a  vessel  sailing  on  a  certain  course,  a  headland 
was  observed  to  bear  due  west ;  four  hours  after  which  it  was 
seen  at  west-southwest ;  and  six  hours  after  this,  the  vessel  con- 
tinuing to  run  at  the  same  rate,  its  bearing  was  found  to  be 
south-southwest.  What  was  the  vessel's  course  at  the  time  ? 

Ans.  N.  42°  35'  W. 

Prob.  8.  Two  ships  of  war,  intending  to  cannonade  a  fort, 
are,  by  the  shallowness  of  the  water,  kept  so  far  from  it  that  they 
suspect  their  guns  can  not  reach  it  with  effect.  In  order,  there- 
fore, to  measure  the  distance,  they  separate  from  each  other  500 


190  TRIGONOMETRY. 

rods;  then  each  ship  observes  the  angle  which  the  other  ship 
and  the  fort  subtend,  which  angles  are  38°  16X  and  37°  9'. 
What,  then,  is  the  distance  between  each  ship  and  the  fort  ? 

Ans.  312  rods  and  320  rods. 

Prob.  9.  A  ship  from  the  latitude  42°  18'  N.,  sails  southwest 
by  south  until  her  latitude  is  40°  18'  N.  "What  direct  distance 
has  she  sailed,  and  how  many  miles  has  she  sailed  to  the  west- 
ward? 

Ans.  Distance  run  144.3  miles,  and  has  sailed  to  west- 
ward 80.2  miles. 

Prob.  10.  A  ship  having  run  due  east  for  three  days,  at  the 
rate  of  eight  knots  an  hour,  finds  she  has  altered  her  longitude 
15  degrees.  What  parallel  of  latitude  did  she  sail  on  ? 

Ans.  Latitude  50°  12'. 

Prob.  11.  A  ship  in  latitude  43°  30'  N.,  and  longitude  44° 
W.,  sails  southeasterly  532  miles,  until  her  departure  from  the 
meridian  is  420  miles.  Required  the  course  steered,  and  the 
latitude  and  longitude  of  the  ship. 

Ans.  Course  S.  52°  8'  E.,  latitude,  38°  3/5  N., 

longitude  34°  45'  W. 

Prob.  12.  A  ship  from  latitude  43°  20'  N.,  and  longitude  52° 
W.,  sails  E.S.E.  until  her  departure  is  745  miles.     Required 
the  distanca  sailed,  and  the  latitude  and  longitude  of  the  ship. 
Ans.  Distance  808.4  miles,  latitude  38°  11/5  N., 

longitude  35°  35'  A7. 

Prob.  13.  If  the  height  of  the  mountain  called  the  Peak  of 
Teneriffe  be  4  miles,  and  the  angle  taken  at  the  top  of  it,  as 
formed  between  a  plumb-line  and  a  line  conceived  to  touch  the 
earth  in  the  horizon,  or  farthest  visible  point,  be  87°  25'  55X/,  it 
is  required  from  hence  to  determine  the  magnitude  of  the  whole 
earth,  and  the  utmost  distance  that  can  be  seen  on  its  surface 
from  the  top  of  the  mountain,  supposing  the  earth  to  be  a  per- 
fect sphere. 

Ans.  Distance  178.458  miles,  diameter  7957.793  miles. 
Prob.  14.  Required  the  course  and  distance  from  St.  Jago, 
one  of  the  Cape  Verd  islands,  in  latitude  14°  56/  N.,  to  the  island 
of  St. Helena,  in  latitude  15°  45''  S.,  their  difference  of  longitude 
being  30°  12'. 

Ans.  Course  S.  44°  12'  E.,  distance  2567.8  miles, 


EXAMPLES   FOR   PRACTICE.  191 

Prob.  15.  A  ship  from  the  latitude  of  49°  57X  K,  and  longi- 
tude of  30°  W.,  sails  S.  39°  W.,  till  she  arrives  in  the  latitude 
of  45°  31X  N.  Required  the  distance  run,  and  the  longitude  of 
the  ship. 

Ana.  Distance  342.3  miles,  longitude  of  ship  35°  21X  "W. 
Prob.  16.  Find  the  bearing  and  distance  from  San  Francisco, 
latitude  37°  48X  N.,  longitude  122°  28'  W.,  to  Jeddo,  latitude 
35°  40X  N.,  longitude  139°  40/  E.,  by  Mercator's  sailing. 

Ans.  Course  S.  88°  26X  W.,  distance  4705  miles. 
Prob.  17.  Find  the  bearing  and  distance  from  San  Francisco 
to  Batavia  in  Java,  latitude  6°  9X  S.,  longitude  106°  53X  E.,  by 
Mercator's  sailing. 

Ans.  Course  S.  70°  12X  W.,  distance  7783  miles. 
Prob.  18.  Find  the  bearing  and  distance  from  San  Francisco 
to  Port  Jackson,  latitude  33°  51X  S.,  longitude  151°  14X  E.,  by 
Mercator's  sailing. 

Ans.  Course  S.  48°  18X  W.,  distance  6462  miles. 
Prob.  19.  Find  the  bearing  and  distance  from  San  Francisco 
to  Otaheite,  latitude  17°  29X  S.,  longitude  149°  29'  W.,  by  Mer- 
cator's sailing. 

Ans.  Course  S.  24°  44X  W.,  distance  3652  miles. 
Prob.  20.  Find  the  bearing  and  distance  from  San  Francisco 
to  Valparaiso,  latitude  33°  2X  S.,  longitude  71°  41X  W.,  by  Mer- 
cator's sailing. 

Ans.  Course  S.  33°  47X  E.,  distance  5354  miles. 

SPHERICAL  TRIGONOMETRY. 

Prob.  1.  In  the  right-angled  spherical  triangle  ABC,  there  are 
given  the  angle  C  23°  27X  42/x,  and  the  side  b  10°  39X  40/x. 
Required  the  angle  B,  and  the  sides  a  and  c. 

(a  =  11°  35X49X/. 

Ans.  }  c  =  4°  35X  26XX. 

(B=66°58X    lxx. 

Prob.  2.  In  the  spherical  triangle  ABC,  the  side  BC-900, 
the  side  AB  =  32°  57X  6/x,  and  the  side  AC  =  66°  32X.  Required 
the  angles. 

(A  =  132°    2X44XX. 
Ans.  JE=  42°  56X  12X'. 
(C=  23°49'26X/ 


192  TRIGONOMETRY. 

Prob.  3.  In  the  right-angled  spherical  triangle  ABC,  there 
are  given  the  angle  B  =  47°  54'  20",  and  the  angle  C  =  61°  50' 
29".  Required  the  sides. 

(a  =  61°    4'  56". 
Ans.  )  6  =40°  30' 20". 

(Crr  50°   30'  30". 

Prob.  4.  In  the  spherical  triangle  ABC,  the  side  AC  =  90°, 
the  side  AB  =  H5°  9',  and  the  angle  B  =  10i°  40'.  Required 
the  remaining  parts. 

(  BC  =  113°  IS7    7". 

Ans.  ]  A    =  115°  54' 46". 

(C    =  117°  33' 49". 

Prob.  5.  In  the  spherical  triangle  ABC,  the  angle  A  =130° 
5'  22",  the  angle  0  =  36°  45'  28",  and  the  side  AC  =44°  13' 
45"  Required  the  remaining  parts. 

(AB  =  51°    6' 12". 

Ans.  JBC:=840  14' 29". 

(     B  =  32°26/    6". 

Prob.  6.  In  the  spherical  triangle  ABC,  the  angle  A=33°  15' 
7",  B  =  31°  34'  38",  and  C  =  161°  25' 17".  Required  the  sides. 

(a=  80°    5'    4". 
*  b=  70°  10X  30". 
c  =  145°    5X    2". 

>.  7.  In  the  spherical  triangle  ABC,  the  side  AB  =  112° 
22X  58",  AC  =  52°  39'  4",  and  BC  =  89°  16X  53".  Required 
the  angles. 

(  A=   70°  39X  0". 

Ans.  <  B  =  48°  36X  0". 

(  C  =119°  15X  0". 

Prob.  8.  In  the  spherical  triangle  ABC,  the  side  AB  =  76°  35 
36",  AC  =  50°  10'  30",  and  the  angle  A  =  34°  15'  3".  Re- 
quired the  remaining  parts. 

(B    =  42°  15' 13". 

Ans.  <C    =121°  36' 20". 

U3C=  40°    O'lO". 

Prob.  9.  The  latitudes  of  the  observatories  of  Paris  and  Pekin 
are  48°  50'  14"  N.  and  39°  54X  13"  N.,  and  their  difference  of 
longitude  is  114°  T  30".  What  is  their  distance  ? 

Ans  73°  56X  40". 


EXAMPLES   FOR   PRACTICE.  193 

Prob.  10.  Required  the  course  and  distance  from  New  ^ oik, 
latitude  40°  43'  N.,  longitude  74°  0'  W.,  to  San  Francisco,  lat- 
itude 37°  48"  N.,  longitude  122°  28'  W.,  on  the  shortest  route. 
Ans.  The  course  is  N.  78°  16'  W. 

Distance,  2229.8  nautical  miles. 

Prob.  11.  Required  the  course  and  distance  from  San  Fran- 
cisco, latitude  37°  48'  N.,  longitude  122°  28'  "W.,  to  Jeddo,  in 
latitude  35°  40'  N.,  longitude  139°  40'  E.,  on  the  shortest 
route.  Ans.  The  course  is  N.  56°  41'  W. 

Distance,  4461.9  nautical  miles. 

Prob.  12.  Required  the  course  and  distance  from  San  Fran- 
cisco to  Batavia  in  Java,  latitude  6°  9'  S.,  longitude  106°  53'  E., 
on  the  shortest  route. 

Ans.  The  course  is  N.  67°  30'  W. 

Distance,  7516  nautical  miles. 

Prob.  13.  Required  the  course  and  distance  from  San  Fran- 
cisco to  Port  Jackson,  latitude  33°  51'  S.,  longitude  151°  14'  E., 
on  the  shortest  route. 

Ans.  The  course  is  S.  59°  50'  W. 

Distance,  6444  nautical  miles. 

Prob.  14.  Required  the  course  and  distance  from  San  Fran^ 
cisco  to  Otaheite,  latitude  17°  29'  S.,  longitude  149°  29'  W.,  on 
the  shortest  route. 

Ans.  The  course  is  S.  29°  45'  W. 

Distance,  3650.3  nautical  miles. 

Prob.  15.  Required  the  course  and  distance  from  San  Fran- 
cisco to  Valparaiso,  latitude  33°  2X  S.,  longitude  71°  41  W.,  on 
the  shortest  route. 

Ans.  The  course  is  S.  39°  227  E. 

Distance,  5108.5  nautical  miles. 

Prob.  16.  Suppose  two  ports,  one  in  north  latitude  30°,  and 
the  other  in  north  latitude  40°,  the  difference  of  longitude  be- 
tween them  being  50°.     Required  the  bearing  and  distance 
from  each  of  these  ports  to  an  island  that  lies  in  south  latitude 
18°,  and  which  is  equally  distant  from  both  of  the  said  ports- 
Ans.  Bearing  from  first  port,      S.  40°  52'    9X/  E. 
Bearing  from  second  port,  S.  15      9  47  "VV. 
The  distance,  59°  23' 19"  =  3563.3  nautical  miles. 
N 


194  TEIGONOMETKY. 

Solution  of  Problem  10,  page  183. 
Let  a = angle  ACD,  fig.  p.  68. 
289 
:  2  ' 
AE  =  r  sin.  a. 
CE  =  r  cos.  a. 
circle  =  TT  r2,  Art.  98. 

sector  ACD  =~v  r2,  Art.  99. 

OOU 

triangle  AEC=J  r2  sin.  a  cos.  a. 
perm-segment  AED-J  TT  r2. 

^^^-ir2sin.«cos.a  =  ^^. 

-t  qrv 

a— sin.  a  cos.  a +  60. 

T 

a  =  28.648  sin.  2a  +  60,  Art.  73. 

This  equation  is  readily  solved  by  the  method  of  approxima- 
tion, Algebra,  Art.  470. 


=  139.33  feet.    Ans. 


THE   ENB 


TABLES 


or 


LOGARITHMS  OF  NUMBERS 


AND    OF 


SUES  AID  TiJ.fcES.i8 


FOR   EVERY 


TEN  SECONDS  OF  THE  QUADRANT, 


WITH  OTHER  USEFUL  TABLES. 


BY  ELIAS   LOO  MIS,  LUX, 

^BOFESSOB    OF  NATUBAL    PHILOSOPHY   AND   ASTRONOMY   IN   TALE    COLLEGE,  AND    ATJTHOB    O»  A 
"COUliSE   OV  MATHEMATICS." 


T  II  I  B  T  Y-S  IXTH   EDITION. 


N  E  W  -  Y  O  R,  K  : 

HARPER   &   BROTHERS,   PUBLISHERS, 
329    &    331    PEARL    STREET, 

FRANKLIN   SQUARE. 

1878. 


Entered,  according  to  Act  of  Congress,  in  the  year  one  thousand  eight  hundred  and  £/rty-eJ^ht,  by 

HARPER  &  BROTHERS, 
in  the  Clerk's  Office  of  the  District  Court  of  the  Southern  District  of  New  York. 


CONTENTS. 


Page 

EXPLANATION  OF    THE    TABLES V 

TABLE    OF   LOGARITHMS   OF   NUMBERS 1 

LOGARITHMIC    SINES   AND    TANGENTS 21 

NATURAL    SINES   AND    TANGENTS ,.,  116 

NATURAL    SECANTS ,           134 

LENGTHS    OF    CIRCULAR    ARCS 135 

TRAVERSE    TABLE 130 

MERIDIONAL    PARTS 112 

CORRECTIONS    TO   MIDDLE    LATITUDE 149 

LOGARITHM?   FOR    COMPOUND     INTEREST,    ETC „ .      ,  .  ,  .  150 


P  E  E  F  A  C  E. 


THE  accompanying  tables  were  designed  to  afford  the  means  of  per- 
forming trigonometrical  computations  with  facility  and  precision.  The 
tables  chiefly  used  in  this  country  for  purposes  of  education  extend  lu 
six  decimal  places,  like  those  in  the  present  collection ;  but  the  pre- 
cision which  they  are  designed  to  furnish  is  only  attained  by  a  serious 
expenditure  of  labor.  In  the  Table  of  Logarithms  of  Numbers  they  do 
not  furnish  the  correction  for  a  fifth  figure  in  the  natural  number,  and 
the  labor  of  computing  this  correction  is  such  that  I  always  prefer  the 
use  of  Hutton's  Tables,  extending  to  seven  places,  even  in  computations 
to  which  six-place  logarithms  are  abundantly  competent.  In  the  pres- 
ent collection,  the  correction  for  a  fifth  figure  of  the  natural  number  is 
introduced  at  the  bottom  of  each  page,  and  the  table  is  thus  rendered 
nearly  as  useful  as  one  of  the  common  kind  extending  to  100,000.  The 
whole  has  been  carefully  compared  with  standard  authors,  and  nearly 
a  dozen  errors  have  thus  been  detected  in  the  common  tables. 

The  principal  table  in  this  collection  is  that  of  Logarithmic  Sines  and 
Tangents.  The  common  tables  in  this  country  extend  only  to  minute* 
with  differences  to  100".  If,  in  a  trigonometrical  computation,  angles 
are  only  required  to  the  nearest,  minute,  tables  to  five  places  are  quite 
sufficient ;  but  if  the  computation  is  to  be  carried  to  seconds,  these  can 
only  be  obtained  from  the  common  tables  by  a  great  expenditure  of 
time  and  labor.  In  the  present  collection,  the  sines  and  tangents  are  fur- 
nished to  every  ten  seconds  of  the  quadrant,  and  at  the  bottom  of  each 
pa§e  is  given  the  correction  for  any  number  of  seconds  less  than  ten,  so 
that  the  precision  of  seconds  can  be  obtained  with  almost  the  same  fa- 
cility as  that  of  minutes-  with  the  tables  in  common  use.  Moreover, 
near  the  limits  of  the  quadrant,  by  means  of  an  auxiliary  table,  sines  and 
tangents  are  readily  obtained,  even  for  a  fraction  of  a  second.  The 
method  of  arrangement  of  the  sines  and  tangents  was  suggested  by  a 
table  in  Mackay's  Longitude  ;  but  the  errors  of  that  table,  amounting  to 
several  thousand,  have  been  corrected  by  a  careful  comparison  with  the 
work  of  Ursinus.  By  comparison  with  the  same  standard,  more  than 
two  hundred  errors  (chiefly  in  the  final  figures)  have  been  detected  in 
the  tables  in  common  use. 

The  Table  of  Natural  Sines  and  Tangents  is  of  less  use  than  the  loga- 
rithmic ;  nevertheless,  it  is  often  important  for  reference,  particularly  in 
analytical  geometry  and  the  calculus  ;  and  it  is  useful  as  a  stepping- 
cfone  to  assist  the  beginner  in  comprehending  the  nature  of  logarithmic 


LV  PREFACE. 

sines  and  tangents.  The  Traverse  Table  commonly  used  in  this  country 
furnishes  the  latitude  and  departure  to  every  quarter  degree  of  the  quad- 
rant, for  distances  from  1  to  100,  and  occupies  ninety  pages.  The  accom- 
panying table  occupies  but  six  pages,  and  yields  ten  times  greater  pre- 
cision. 

The  Table  of  Meridional  Parts  extends  to  tenths  of  a  mile,  and  great 
care  has  been  taken  to  insure  its  accuracy.  For  this  purpose,  I  have 
compared  all  the  similar  tables  within  my  reach,  and  among  them  have 
found  two  which  appeared  to  have  been  computed  independently.  Be- 
tween them  there  were  detected  674  discrepancies  in  the  final  figures. 
These  cases  were  all  recomputed,  and  78  errors  were  detected  in  the 
best  copy  compared.  It  is  probable  that  the  numbers  in  this  table  are 
not  in  every  instance  true  to  the  nearest  tenth  of  a  mile  ;  but  it  is  be- 
lieved that  the  remaining  errors  are  few  in  number,  as  well  as  minute. 
This  table  is  confidently  pronounced  more  accurate  than  any  similar 
one  with  which  I  have  been  able  to  compare  it. 

The  Table  of  Corrections  to  Middle  Latitude  was  computed  entirely 
anew.  The  corresponding  table  in  common  use,  which  was  originally 
computed  by  Workman,  contains  more  than  four  hundred  errors,  sev- 
eral of  them  amounting  to  two  minutes. 

On  the  whole,  it  is  believed  that  the  accompanying  tables  will  be 
found  more  convenient  to  the  computer  than  any  tables  of  six  decimal 
places  hitherto  published  in  this  country ;  and  that  they  will  be  pro- 
nounced sufficiently  extensive  for  all  purposes  of  academic  and  coliegi- 
ate  instruction,  as  well  as  for  practical  mechanics  and  surveyors. 


EXPLANATION  OF  THE  TABLES, 


TABLE  OF  LOGARITHMS  OF  NUMBERS,  pp.  1-20. 

LOGARITHMS  are  numbers  contrived  to  diminish  the  labor  of  Multiplica- 
tion and  Division  by  substituting  in  their  stead  Addition  and  Subtrac- 
tion. All  numbers  are  regarded  as  powers  of  some  one  number,  which 
Is  called  the  base  of  the  system  ;  and  the  exponent  of  that  power  of  the 
base  which  is  equal  to  a  given  number,  is  called  the  logarithm  of  that 
number. 

The  base  of  the  common  system  of  logarithms  (called,  from  their  in- 
ventor, Briggs'  logarithms)  is  the  number  10.  Hence  all  numbers  are 
to  be  regarded  as  powers  of  10.  Thus,  since 

10°— 1,  0  is  the  logarithm  of  1  in  Briggs'  system  ; 

10'=  10,          1     "  "  10  "  " 

10'=  100,        2     "  "  100  "  " 

10'=  1000,      3     "  "  1000  "  " 

10*=  10,000,  4     "  "  10,000  "  " 

&c.,  &c.,  &c. ; 

whence  it.  appears  that,  in  Briggs'  system,  the  logarithm  of  every  num- 
ber between  1  and  10  is  some  number  between  0  and  1,  i.  e.,  is  a  prop- 
er fraction.  The  logarithm  of  every  number  between  10  and  100  is 
some  number  between  1  and  2,  i.  e.,  is  1  plus  a  fraction.  The  loga- 
rithm of  every  number  between  100  and  1000  is  some  number  between 
2  and  3,  i.  e.,  is  2  plus  a  fraction,  and  so  on. 

The  preceding  principles  may  be  extended  to  fractions  by  means  of 
negative  exponents.  Thus,  since 

10"1— -0.1,         — 1  is  the  logarithm  of  0.1  in  Briggs'  system 
10-*=0.01,       —2       "                  "           0.01 
10~3=0.001,     —3       "                  "           0.001  "  " 

10~4=0.0001,  —4       "  "          0.0001  "  " 

&c.,  &c.,  &c. 

Hence  it  appears  that  the  logarithm  of  every  number  between  1  and 
0.1  is  some  number  between  0  and  — 1,  or  may  be  represented  by  — 1 
plus  a  fraction;  the  logarithm  of  every  number  between  0.1  and  .01  is 
some  number  between  — 1  and  — 2,  or  may  be  represented  by  —  2  plus 


v  i  EXPLANATION    OF    THE    TABLES. 

a  fraction  ;  the  logarithm  of  every  number  between  .01  and  .001  is  seme 
number  between  — 2  and  — 3,  or  is  equal  to  — 3  plus  a  fraction  and  so  on, 
The  logarithms  of  most,  numbers,  therefore,  consist  of  an  integer  and 
it  fraction.  The  integral  part  is  called  the  characteristic,  and  may  be 
known  from  the  following 

RULE. 

The  characteristic  of  the  logarithm  of  a  number  greater  than  unity,  «•>• 
one  less  than  the  number  of  integral  figures  in  the  given  number. 

Thus  the  logarithm  of  297  is  2  plus  a  fraction  ;  that  is,  the  character- 
istic of  the  logarithm  of  297  is  2,  which  is  one  less  than  the  number  ot 
integral  figures.  The  characteristic  of  the  logarithm  of  5673.29  is  3  ; 
ihat  of  73254.1  is  4,  &c. 

The  characteristic  of  the  logarithm  of  a  decimal  fraction  is  a  negative 
number,  and  is  equal  to  the  number  of  places  by  which  its  first  significant 
figure  is  removed  from  the  place  of  units. 

Thus  the  logarithm  of  .0046  is  — 3  plus  a  fraction  ;  that  is,  the  char- 
acteristic of  the  logarithm  is  — 3,  the  first  significant  figure  4  being 
removed  three  places  from  units. 

The  accompanying  table  contains  the  logarithms  of  all  numbers  from 
1  to  10,000  carried  to  6  decimal  places. 

To  find  the  Logarithm  of  any  Number  between  1  and  100. 

Look  on  the  first  page  of  the  table,  along  the  column  of  numbers  under 
N,  for  the  given  number,  and  against  it,  in  the  next  column,  will  be  found 
the  logarithm,  with  its  characteristic.  Thus, 

opposite  13  is  1.113943,  which  is  the  logarithm  of  13; 
65  is  1.812913,  "  "  65. 

To  find  i.ht  Logarithm  of  any  Number  consisting  of  three  Figures. 

Look  on  one  of  the  pages  from  2  to  20,  along  the  left-hand  column 
marked  N,  for  the  given  number,  and  against  it,  in  the  column  headed  0, 
will  be  found  the  decimal  part  of  its  logarithm.     To  this  the  character- 
stic   rnnst  be  prefixed,  according  to  the  rule  already  given.     Thus 
the  logarithm  of  347  will  be  found,  from  page  8,  to  be  2.540329 ; 
"  "     871          "  "  "      18,    "     2.940018. 

As  the  fiist  two  figures  of  the  decimal  are  the  same  for  several  suc- 
cessive numbers  in  the  table,  they  are  not  repeated  for  each  logarithm 
separately,  but  are  left  to  be  supplied.  Thus  the  decimal  part  of  th« 
logarithm  of  339  is  .530200.  The  first  two  figures  of  the  decimal  remain 
the  same  up  to  347 ;  they  are  therefore  omitted  in  the  table,  and  are  to 
be  supplied. 

To  find  the  Logarithm  of  any  Number  consisting  of  four  Figures. 
Find  the  three  left-hand  figures  in  the  column  marked  N  as  before, 


EXPLANATION   OF   THE   TABLES.  vii 

and  the  fourth  figure  at  the  head  of  one  of  the  other  columns.     Opposite 
to  the  first  three  figures,  and  in  the  column  under  the  fourth  figure,  will 
he  found  four  figures  of  the  logarithm,  to  which  two  figures  from  the 
column  headed  0  are  to  be  prefixed,  as  in  the  former  case.     The  char- 
acteristic must  be  supplied  by  the  usual  rule.     Thus 
the  logarithm  of  3456  is  3.538574 ; 
"     8765  is  3.942752. 

In  several  of  the  columns  headed  1,  2,  3,  &c.,  small  dots  are  found  in 
the  place  of  figures.  This  is  to  show  that  the  two  figures  which  are  to 
be  prefixed  from  the  first  column  have  changed,  and  they  are  to  be 
taken  from  the  horizontal  line  directly  below.  The  place  of  the  dots  is 
to  be  supplied  with  ciphers.  Thus 

the  logarithm  of  2045  is  3.310693  ; 
"  "     9777  is  3.990206. 

The  two  leading  figures  from  the  column  0  must  also  be  taken  from 
the  horizontal  line  below,  if  any  dots  have  been  passed  over  on  the  same 
horizontal  line.  Thus 

the  logarithm  of  1628  is  3.211654. 

To  find  the  Logarithm  of  any  Number  containing  more  than  four 

Figures. 

By  inspecting  the  table,  we  shall  find  that  within  certain  limits  the  log- 
arithms are  nearly  proportional  to  their  corresponding  numbers.     Thus 
the  logarithm  of  7250  is  3.860338  ; 
"     7251  is  3.860398  ; 
"     7252  is  3.860458  ; 
"     7253  is  3.860518. 

Here  the  difference  between  the  successive  logarithms,  called  the 
tabular  difference,  is  constantly  60,  corresponding  to  a  difference  of  unity 
in  the  natural  numbers.  If,  then,  we  suppose  the  logarithms  to  be  pro- 
portional to  their  corresponding  numbers  (as  they  are  nearly),  a  differ- 
ence of  0.1  in  the  numbers  should  correspond  to  a  difference  of  6  in  the 
logarithms  ;  a  difference  of  0.2  in  the  numbers  should  correspond  to  a 
difference  of  12  in  the  logarithms,  &c.  Hence 

the  logarithm  of  7250.1  must  be  3.860344; 

"     7250.2         "       3.860350; 

"  "     7250.3         "       3.860356; 

&c.,  &c. 

in  order  to  facilitate  the  computation,  the  tabular  difference  is  insert- 
ed on  page  16  in  the  column  headed  D,  and  the  proportional  part  for  the 
fifth  figure  of  the  natural  number  is  given  at  the  bottom  of  the  page, 
Thus,  when  the  tabular  difference  is  60,  the  corrections  for  .1,  .2,  .3; 
&c.,  are  seen  to  be  6,  12,  18,  &c. 

If  the  given  number  was  72501.  the  characteristic  of  its  logarithm 
would  be  4,  but  the  decimal  part  would  be  the  same  as  for  7250.1. 


EXPLANATION    OF    THE    TABLES. 

If  it  were  required  to  find  the  correction  for  a  sixth  figure  in  the  nat- 
ural number,  it  is  readily  obtained  from  the  Proportional  Parts  in  the 
table.  Thus,  if  the  correction  for  .5  is  30,  the  correction  for  .05  is  ob- 
viously 3. 

As  the  differences  change  rapidly  in  the  first  part  of  the  table,  it 
was  found  inconvenient  to  give  the  proportional  parts  for  each  tabular 
difference ;  accordingly,  for  the  first  seven  pages  they  are  only  given 
for  the  even  differences,  but  the  proportional  parts  for  the  odd  differences 
will  be  readily  found  by  inspection. 
Required  the  logarithm  of  452789. 

The  logarithm  of  452700  is  5.655810 
The  tabular  difference  is  96. 

Accordingly,  the  correction  for  the  fifth  figure,  8,  is  77,  and  for  the 
sixth  figure,  9,  is  8.6,  or  9  nearly.  Adding  these  corrections  to  the  num- 
ber before  found,  we  obtain  5.655896. 

The  preceding  logarithms  do  not  pretend  to  be  perfectly  exact,  but 
only  the  nearest  numbers  having  but  six  decimal  places.     Accordingly, 
when  the  fraction  which  is  omitted  exceeds  half  a  unit  in  the  sixth  deci- 
mal place,  the  last  figure  must  be  increased  by  unity. 
Required  the  logarithm  of  8765432. 

The  logarithm  of  8765000  is  6.942752 

Correction  for  the  fifth  figure  4,  20 

"       sixth  figure  3,  1.5 

"  "       seventh  figure  2,  0.1 

Therefore  the  logarithm  of  8765432  is  6.942774. 

Required  the  logarithm  of  234567.  . 

The  logarithm  of  234500  is  5.370143 

Correction  for  the  fifth  figure  6,  111 

"               "       sixth  figure  7,  13 

Therefore  the  logarithm  of  234567  is  5.370267. 

To  find  the  Logarithm  of  a  Decimal  Fraction. 

The  decimal  part  of  the  logarithm  of  any  number  is  the  same  as  that 
of  the  number  multiplied  or  divided  by  10,  100,  1000,  &c.  Hence,  for 
a  decimal  fraction,  we  find  the  logarithm  as  if  the  figures  were  integers, 
and  prefix  the  characteristic  according  to  the  usual  rule. 

EXAMPLES. 

The  logarithm  of  345.6  is  2.538574 ; 

"  «        87.65  is  1.942752; 

«  «          2.345  is  0.370143; 

«       "     .1234  is  L091315; 

«     .005678  is  3.754195. 


EXPLANATION    OF    THE    TABLES  ix 

The  minus  sign  is  placed  over  the  characteristic  to  show  that  this 
alone  is  negative,  while  the  decimal  part  of  the  logarithm  is  positive. 

To  find  the  Logarithm  of  a  Vulgar  Fraction. 

Wo  may  reduce  the  vulgar  fraction  to  a  decimal,  and  find  its  loga- 
rithm by  the  preceding  rule  ;  or,  since  the  value  of  a  fraction  is  equal  to 
the  quotient  of  the  numerator  divided  by  the  denominator,  we  may  sub- 
tract the  logarithm  of  the  denominator  from  that  of  the  numerator;  the 
difference  will  be  the  logarithm  of  the  quotient. 
Required  the  logarithm  of  T3F,  or  0.1875. 

From  the  logarithm  of  3,          0.477121, 
Subtract  the  logarithm  of  16,  1.204120. 
Leaves  logarithm  of  T3F,  or  .1875,  f. 273001. 

Tn  the  same  manner  we  find 

the  logarithm  of  /T  is  2.861697; 
"  "     jf£  is  1.147401. 

To  find  the  natural  Number  corresponding  to  any  Logarithm. 

Look  in  the  table  in  the  column  headed  0  for  the  first  two  figures  of 
the  logarithm,  neglecting  the  characteristic ;  the  other  four  figures  ara 
to  be  looked  for  in  the  same  column,  or  in  one  of  the  nine  following  col- 
umns ;  and  if  they  are  exactly  found,  the  first  three  figures  of  the  corre- 
sponding number  will  be  found  opposite  to  them  in  the  column  headed 
N,  and  the  fourth  figure  will  be  found  at  the  top  of  the  page.  This  num- 
ber must  be  made  to  correspond  with  the  characteristic  of  the  giver, 
logarithm  by  pointing  off  decimals  or  annexing  ciphers.  Thus 

the  natural  number  belonging  to  the  logarithm  4.370143  is  23450  ; 
"  "  "  "          1.538574  is  34.56. 

If  the  decimal  part  of  the  logarithm  can  not  be  exactly  found  in  the 
table,  look  for  the  nearest  less  logarithm,  and  take  out  the  four  figures 
of  the  corresponding  natural  number  as  before ;  the  additional  figures 
may  be  obtained  by  means  of  the  Proportional  Parts  at  the  bottom  of 
the  page. 

Required  the  number  belonging  to  the  logarithm  4.368399. 

On  page  6,  we  find  the  next  less  logarithm  .368287. 

The  four  corresponding  figures  of  the  natural  number  are  2335. 
Their  logarithm  is  Jess  than  the  one  proposed  by  112.  The  tabular 
difference  is  186;  and,  by  referring  to  the  bottom  of  page  6,  we  find 
that,  with  a  difference  of  186,  the  figure  corresponding  to  the  Piopor- 
tional  Part  112  is  6.  Hence  the  five  figures  of  the  natural  number  are 
23356  ;  and,  since  the  characteristic  of  the  proposed  logarithm  is  4,  these 
five  figures  are  all  integral. 

Required  the  number  belonging  to  logarithm  5.345678. 

The  next  less  logarithm  in  the  table  is  .345570 

Their  difference  is  108. 


x  EXPLANATION    OF    THE    TABLES. 

The  first  four  figures  of  the  natural  numljer  are  2216. 
With  the  tabular  difference  196,  the  fifth  figure  corresponding  to  108 
s  seen  tc  be  5,  with  a  remainder  of  10,  which  furnishes  a  sixth  figure  5 
nearly.     Hence  the  required  number  is  221655. 
In  the  same  manner  we  find 

the  number  corresponding  to  logarithm  3.538672  is  3456.78 ; 

1.994605  is  98.7654; 
"  "  "  T.647817  is  .444444. 

TABLE  OF  NATURAL  SINES  AND  TANGENTS,  pp.  116—188. 

This  is  a  table  of  natural  sines  and  tangents  for  every  degree  and 
minute  of  the  quadrant,  carried  to  six  places  of  figures.  Since  the  ra- 
dius of  the  circle  is  supposed  to  be  unity,  the  sine  of  every  arc  below 
90°  is  less  than  unity.  These  sines  are  expressed  in  decimal  parts  of  the 
radius ;  and,  although  the  decimal  point  is  not  written  in  the  table,  it 
must  always  be  prefixed.  The  degrees  are  arranged  in  order  at  the  top 
of  the  page,  and  the  minutes  in  the  left  hand  vertical  column.  Directly 
under  the  given  number  of  degrees  at  the  top  of  the  page,  and  opposite 
to  the  minutes  on  the  left,  will  be  found  the  sine  required.  The  twn 
leading  figures  are  repeated  at  intervals  often  minutes.  Thus 
the  sine  of  6°  27'  is  .112336; 
«  28°  53'  is  .483028. 

The  same  number  in  the  table  is  both  the  sine  of  an  arc  and  the  co- 
sirie  of  its  complement.     The  degrees  for  the  cosines  must  be  sought  at 
the  bottom  of  the  page,  and  the  minutes  on  the  right.     Thus 
the  cosine  of  62°  25'  is  .463038 ; 
"  "  84°  23'  is  .097872. 

Ji  a  sine  is  required  for  an  arc  consisting  of  degrees,  minutes,  and  sec- 
onds, it  may  be  found  by  means  of  the  line  at  the  bottom  of  each  page, 
which  gives  the  proportional  part  corresponding  to  one  second  of  arc. 

Required  the  sine  of  8°  9'  10". 

The  sine  of  8°  9'  is  .141765. 

By  leferring  to  the  bottom  of  page  116,  in  the  column  under  8°,  we 
find  the  correction  for  I"  is  4.80  ;  hence  the  correction  for  10"  must  be 
48,  which,  added  to  the  number  above  found,  gives  for  the  sine  of  8° 
9'  10",  .141813. 

In  the  same  manner  we  find 

the  cosine  of  56°  34'  28"  is  .550853. 

It  will  be  observed,  that  since  the  cosines  decrease  while  the  arcs  in- 
crease, the  correction  for  the  28"  is  to  be  subtracted  from  the  cosine 
of  56°  34'. 

The  arrangement  of  the  table  of  natural  tangents  is  similar  to  that  of 
the  table  of  sines.  The  tangents  for  arcs  less  than  45°.  are  all  less  than 
radius,  and  consist  wholly  of  decimals.  For  arcs  above  45°,  the  tan- 
gents are  all  gi eater  than  radius  and  contain  both  integral  and  decima. 


EXPLANATION    OF    THE    T  A  D  L K s.  x i 

figures.     The  proportional  parts  at  the  bottom  of  each  page  enable  us 
readily  to  find  the  correction  for  seconds.     Thus 

the  natural  tangent  of  32°  29'  IS"  is  .636784  ; 
"      743  35'  55"  is  3.63014. 

To  find  the  Number  of  Degrees,  Minutes,  and  Seconds  belonging  to  a 
given  Sine  or  Tangent. 

If  the  given  sine  or  tangent  is  found  exactly  in  the  table,  the  corre- 
sponding degrees  will  be  found  at  the  top  of  the  page,  and  the  minutes 
on  the  left  hand.  But  when  the  given  number  is  not  found  exactly  ii) 
the  table,  look  for  the  sine  or  tangent  which  is  next  less  than  the  pro- 
posed one,  and  take  out  the  corresponding  degrees  and  minutes.  Find, 
also,  the  difference  between  this  tabular  number  and  the  number  pro- 
posed, and  divide  it  by  the  proportional  part  for  I"  found  at  the  bottom 
of  the  page ;  the  quotient  will  be  the  required  number  of  seconds. 

Required  the  arc  whose  sine  is  .750000. 

The  next  less  sine  in  the  table  is  .749919,  the  arc  corresponding  tu 
which  is  48°  35'.  The  difference  between  this  sine  and  that  proposed 
is  81,  which,  divided  by  3.21,  gives  25.  Hence  the  required  arc  is  4SG 
35'  25". 

Tn  the  same  manner  we  find 

the  arc  whose  tangent  is  2.000000,  to  be  63°  26'  6". 

TABLE  OF  NATURAL  SECANTS,  pp.  134-5. 

This  is  a  table  of  natural  secants  for  every  ten  minutes  of  the  quad- 
rant carried  to  seven  places  of  figures.  The  degrees  are  arranged  in 
order  in  the  first  vertical  column  on  the  left,  and  the  minutes  at  the  top 
of  the  page.  Thus 

the  secant  of  21°  20'  is  1.073561  ; 
"    81°  50'  is  7.039622. 

If  a  secant  is  required  for  a  number  of  minutes  not  given  in  the  table, 
the  correction  for  the  odd  minutes  may  be  found  by  means  of  the  last 
vertical  column  on  the  right,  which  shows  the  proportional  part  for  one 
minute. 

Let  it  be  required  to  find  the  secant  of  30°  33'. 

The  secant  of  30°  30'  is  1.160592. 

The  correction  for  1'  is  198.9,  which,  multiplied  by  3,  gives  597. 
Adding  this  to  the  number  before  found,  we  obtain  1.161189. 

For  a  cosecant,  the  degrees  must  be  sought  in  the  right-hand  vertical 
column,  and  the  minutes  at  the  bottom  of  the  page.     Thus 
the  cosecant  of  47°  40'  is  1.352742. 

TABLE  OF  LOGARITHMIC  SINES  AND  TANGENTS,  pp.  21—115. 

This  is  a  table  of  the  logarithms  of  the  sines  and  tangents  for  every 
ten  seconds  of  the  quadrant,  carried  to  six  places  of  decimals  The  de« 


xii  EXPLANATION    OF   THE    TABLES. 

grees  and  seconds  are  placed  at  the  top  of  the  page,  and  the  minutes  in 
the  left  vertical  column.  After  the  first  two  degrees,  the  three  leading 
figures  in  the  table  of  sines  are  only  given  in  the  column  headed  0",  and 
are  to  be  prefixed  to  the  numbers  in  the  other  columns,  as  in  the  table 
of  logarithms  of  numbers.  Also,  where  the  leading  figures  change,  this 
change  is  indicated  by  dots,  as  in  the  former  table.  The  correction  for 
any  number  of  seconds  less  than  10  is  given  at  the  bottom  of  the  page. 

To  find  the  Logarithmic  Sine  or  Tangent  of  a  given  Arc. 

Look  for  the  degrees  at  the  top  of  the  page,  the  minutes  on  the  left 
hand,  and  the  next  less  tenth  second  at  the  top  ;  then,  under  the  seconds, 
and  opposite  to  the  minutes,  will  be  found  four  figures,  to  which  the 
three  leading  figures  are  to  be  prefixed  from  the  column  headed  0" ;  to 
this  add  the  proportional  part  for  the  odd  seconds  from  the  bottom  of 
the  page. 

Required  the  logarithmic  sine  of  24°  27'  34". 

The  logarithmic  sine  of  24°  27'  30"  is  9.G17033. 
Proportional  part  for  4"  is  18. 

Logarithmic  sine  of  24°  27'  34"  is       9.617051. 

This  is  the  logarithm  of  .414049  found  in  the  table  of  natural  sines  on 
page  120.  The  natural  sine  being  less  than  unity,  the  characteristic 
of  its  logarithm  is  negative.  Tc  obviate  this  inconvenience,  the  char- 
acteristics in  the  table  nave  all  been  increased  by  10;  or  the  logarith- 
mic sines  may  be  regarded  as  the  logarithms  of  natural  sines  computed 
for  a  radius  of  10,000,000,000. 

Required  the  logarithmic  tangent  of  73°  35'  43". 

The  logarithmic  tangent  of  73°  35'  40"  is  10.531031. 
Proportional  part  for  3"  is  23. 

Logarithmic  tangent  of  73°  3.V  43"  10.531054. 

When  a  cosine  is  required,  the  degrees  and  seconds  must  be  sought 
at  the  bottom  of  the  page,  and  the  minutes  on  the  right,  and  the  correc- 
tion for  the  odd  seconds  must  be  subtracted  from  the  number  in  the  table, 
Required  the  logarithmic  cosine  of  59°  33'  47". 

The  logarithmic  cosine  of  59°  33'  40"  is  9.704682. 
Proportional  part  for  7"  is  25. 

Logarithmic  cosine  of  59°  33'  47"  is  9.704657. 
So,  also,  the  logarithmic  cotangent  of  37°  27'  1 4"  ic  found  to  be  10.115744. 
The  proportional  parts  given  at  the  bottom  of  each  page  correspond 
to  the  degrees  at  the  top  ol  the  page  increased  by  30',  and  are  not 
strictly  applicable  to  any  other  number  of  minutes ;  nevertheless,  the 
differences  of  the  sines  change  so  slowly,  except  near  the  commence- 
ment of  the  quadrant,  that  the  error  resulting  from  using  these  numbers 
for  every  part  of  the  page  will  seldom  exceed  a  unit  in  the  sixth  deci- 
mal place.  For  the  ftrsf  two  degrees,  the  differences  change  so  rapidly 


E  X  r  1  A  N  A  T  f  O  N      OF     THE     T  A  B  L  E  S.  X  1 1 1 

that  the  proportional  part  for  1"  is  given  for  each  minute  in  tho  right- 
hand  column  of  the  page.  The  correction  for  any  number  of  seconds 
less  than  ten  will  be  found  by  multiplying  the  proportional  part  for  V 
by  the  given  number  of  seconds. 

Required  the  logarithmic  sine  of  1°  17'  33". 

The  logarithmic  sine  of  1°  17'  30"  is  8.352991. 

The  correction  for  3"  is  found  by  multiplying  93.4  by  3,  which  gives 
280.     Adding  this  to  the  above  tabular  number,  we  obtain 
the  sine  of  1°  17'  33",  8.353271. 

A  similar  method  may  be  employed  for  several  of  the  first  degrees 
of  the  quadrant,  if  the  proportional  parts  at  the  bottom  of  the  page  are 
net  thought  sufficiently  precise.  This  correction  may,  however,  be  ob- 
tained pretty  nearly  by  inspection  from  comparing  the  proportional 
parts  for  two  successive  degrees.  Thus,  on  page  26,  the  correction  for 
1",  corresponding  to  the  sine  of  2°  30',  is  48  ;  the  correction  for  1",  cor- 
responding to  the  sine  of  3°  30',  is  34.  Hence  the  correction  for  1", 
corresponding  to  the  sine  of  3°  0',  must  be  about  41  ;  and  in  the  same 
manner  we  may  proceed  for  any  other  part  of  the  table. 

Near  the  close  of  the  quadrant,  the  tangents  vary  so  rapidly,  that  the 
same  arrangement  of  the  table  is  adopted  as  for  the  commencement  of 
the  quadrant.  For  the  last  as  well  as  the  first  two  degrees  of  the  quad- 
rant, the  proportional  part  to  1"  is  given  for  each  minute  separately. 
These  proportional  parts  are  computed  for  the  minutes  placed  opposite 
to  them,  increased  by  30' ,  and  are  not  strictly  applicable  to  any  other 
number  of  seconds  ;  nevertheless,  the  differences  for  the  most  part 
change  so  slowly,  that  the  error  resulting  from  using  these  numbers  for 
every  part  of  the  same  horizontal  line  is  quite  small.  When  great  ac- 
curacy is  required,  the  table  on  page  114  may  be  employed  for  arcs 
near  the  limits  of  the  quadrant.  This  table  furnishes  the  differences  be* 
tween  the  logarithmic  sines  and  the  logarithms  of  the  arcs  expressed  in 
seconds.  Thus 

the  logarithmic  sine  of  0°  5'  is  7.16269G  ; 
the  logarithm  of  300"  (=5')  is  2.477121  ; 

the  difference  is  4.G85575. 

This  is  the  number  found  on  page  114,  under  the  heading  log.  sine 
A — log.  A",  opposite  5  min. ;  and  in  a  similar  manner  the  other  numbers 
in  the  same  column  are  obtained.  These  numbers  vary  quite  slowly 
for  two  degrees  ;  and  hence,  to  find  the  logarithmic  sine  of  an  arc  less 
than  two  degrees,  we  have  but  to  add  the  logarithm  of  the  arc  expressed 
in  seconds  to  the  appropriate  number  found  in  this  table. 
Required  the  logarithmic  sine  of  0°  7'  22". 

Tabular  number  from  page  114,  4.685575. 
The  logarithm  of  442"  is  2.645422 

Logarithmic  sine  of  0°  7'  22"  is  7.330997. 


xiv  EXPLANATION    OF    THE    TABLES. 

The  logarithmic  tangent  of  an  arc  less  than  two  degrees  is  found  in 
a  sim:.!ar  manner. 

Required  the  logarithmic  tangent  of  0°  21'  36". 

Tabular  number  from  page  114,       4.G85584. 
The  logarithm  of  1656"  is  3.219060. 

Logarithmic  tangent  of  0°  21'  36"  is  7.904644. 

The  column  headed  log.  cot.  A+log.  A"  is  found  by  adding  the  log- 
arithmic cotangent  to  the  logarithm  of  the  arc  expressed  in  seconds. 
Hence,  to  find  the  logarithmic  cotangent  of  an  arc  less  than  two  degrees, 
we  must  subtract  from  the  tabular  number  the  logarithm  of  the  arc  in 
seconds. 

Required  the  logarithmic  cotangent  of  0°  21'  36". 

Tabular  number  from  page  114,  15.314416. 

The  logarithm  of  1656"  is  3.219060. 

Logarithmic  cotangent  of  0°  21'  36"  is  12.095356. 
The  same  method  will,  of  course,  furnish  cosines  and  cotangents  ot 
arcs  near  90°. 

The  secants  and  cosecants  are  omitted  in  this  table,  since  they  arc 
easily  derived  from  the  cosines  and  sines. 

The  logarithmic  secant  is  found  by  subtracting  the  logarithmic  ccsine 
from  20  ;  and  the  logarithmic  cosecant  is  found  by  subtracting  the  loga- 
rithmic sine  from  20. 

Thus  we  have  found  the  logarithmic  sine  of  24°  21'  34"  to  be  y.617051. 
Hence  the  logarithmic  cosecant  of  24°  27'  34"  is  10.382949. 

The  logarithmic  cosine  of  54°  12'  40"  is  9.767008. 

Hence  the  logarithmic  secant  of  54°  12'  40"  is  10.232992. 

To  find  the  Arc  corresponding  to  a  given  Logarithmic  Sine  or  Tangent. 

If  the  given  number  is  found  exactly  in  the  table  the  corresponding 
degrees  and  seconds  will  be  found  at  the  top  of  the  page,  and  the  min- 
utes on  the  left.  But  when  the  given  number  is  not  found  exactly  in 
the  table,  look  for  the  sine  or  tangent  which  is  next  less  than  the  pro- 
posed one,  and  take  out  the  corresponding  degrees,  minutes,  and  sec 
onds.  Find,  also,  the  difference  between  this  tabular  number  and  the 
number  proposed,  and,  corresponding  to  this  difference,  at  the  bottom  of 
the  page  will  be  found  a  certain  number  of  seconds,  which  is  to  be  added 
to  the  arc  before  found. 

Required  the  arc  corresponding  to  the  logarithmic  sine  9.750000. 

The  next  less  sine  in  the  table  is  9.749987. 

The  arc  corresponding  to  which  is  34°  13'  0". 

The  difference  between  its  sine  and  the  one  proposed  is  13,  corre- 
sponding to  which,  at  the  bottom  of  the  page,  we  find  4"  nearly,  Hence 
the  required  arc  is  34°  13'  4". 


EXPLANATION  OF  THE  TABLES,          xv 

In  the  same  manner  we  find  the  arc  corresponding  to  logarithmic  tan- 
gent 10.250000,  to  be  60°  38'  57". 

When  the  arc  falls  within  the  first  two  degrees  of  the  quadrant,  the 
odd  seconds  may  be  found  by  dividing  the  difference  between  the  tab- 
ular number  and  the  one  proposed,  by  the  proportional  part  for  1".  We 
thus  find  the  arc  corresponding  to  logarithmic  sine  8.400000,  to  he  1° 
26'  22"  nearly. 

We  may  employ  the  same  method  for  the  last  two  degrees  of  the 
quadrant  when  a  tangent  is  given ;  but  near  the  limits  of  the  quadrant 
it  is  better  to  employ  the  auxiliary  table  on  page  114.  If  wre  subtract 
the  corresponding  tabular  number  on  page  114  from  the  given  logarith- 
mic sine,  the  remainder  will  be  the  logarithm  of  the  arc  expressed  in 
seconds. 

Required  the  arc  corresponding  to  logarithmic  sine  7.000000. 

We  see,  from  page  22,  that  the  arc  must  be  nearly  3'  ;  the  correspond- 
ing tabular  number  on  page  114  is  4.685575. 

The  difference  is  2.314425  ; 
which  is  the  logarithm  of  206."265. 

Hence  the  required  arc  is  3'  26."265. 

In  the  same  manner  we  find  the  arc  corresponding  to  logarithmic 
tangent  8.184008,  to  be  0°  52'  35". 

TABLE  FOR  THE  LENGTHS  OF  CIRCULAR  ARCS,  p.  135. 

This  table  contains  the  lengths  of  every  single  degree  up  to  60,  and 
at  intervals  often  degrees  up  to  180  ;  also  for  every  minute  and  second 
up  to  20.  The  lengths  are  all  expressed  in  decimal  parts  of  radius. 

Required  the  length  of  an  arc  of  57°  17'  44."8. 

Take  out  from  their  respective  columns  the  lengths  answering  to  each 
of  these  numbers  singly,  and  add  them  all  together  thus : 


57°  .  .  . 

.  .  .  0.9948377 

17'  ... 

.  .  .   .0049451 

40"  .  .  . 

.  .     .0001939 

4"  .  .  . 

.  .  .   .0000194 

0."8  .  .  . 

.  .  .   .0000039 

The  sum  is         1.0000000. 

That  is,  the  length  of  an  arc  of  57°  17'  44. "8  is  equal  to  the  radius  of 
the  circle. 

TRAVERSE  TABLE,  pp.  136-141. 

This  table  shows  the  difference  of  latitude  and  the  departure  to  four 
decimal  places  for  distances  from  1  to  10,  and  for  bearings  from  0°  to 
00°,  at  intervals  of  15'.  If  the  bearing  is  less  than  45°,  the  angle  will 
be  found  on  the  left  margin  of  one  of  the  pages  of  the  table,  and  the  dis- 
tance at  the  top  or  bottom  of  the  page  ;  the  difference  of  latitude  will 


xvi  EXPLANATION    OF    THE    TABLES. 

be  found  in  the  column  headed  lat.  at  the  top  of  the  page,  and  the  de- 
parture in  the  column  headed  dep.  If  the  bearing  is  more  than  45°,  the 
angle  will  be  found  on  the  right  margin,  and  the  difference  of  latitude 
will  bo  found  in  the  column  marked  lat.  at  the  bottom  of  the  page,  and 
the  departure  in  the  other  column.  The  latitudes  and  departures  for 
different  distances  with  the  same  bearing,  are  proportional  to  the  dis- 
tances. Therefore  the  distances  may  be  reckoned  as  tens,  hundreds,  or 
thousands,  if  the  place  of  the  decimal  point  in  each  departure  and  differ- 
ence of  latitude  be  changed  accordingly. 

Required  the  latitude  and  departure  for  the  distance  32.25,  and  the 
bearing  10°  30'. 

On  page  136,  opposite  to  10°  30',  we  find  the  following  latitudes  and 
departures,  proper  attention  being  paid  to  the  position  of  the  decimal 
points. 

Distance.  Diff.  Lat.  DC  p. 

30  29.498  5.467 

2  1.966  .364 

.2  19?  .036 

.05  .049  .009 

32.25  SlTnO  57876. 

TABLE  OF  MERIDIONAL   PARTS,  pp.  142-148 

This  table  gives  the  length  of  the  enlarged  meridian  on  Mercator's 
Chart  to  every  minute  of  latitude  expressed  in  geographical  miles  and 
tenths  of  a  mile.     The  degrees  of  latitude  are  arranged  in  order  at  the 
top  of  the  page,  and  the  minutes  on  both  the  right  and  left  margins. 
Under  the  degrees  and  opposite  to  the  minutes  are  placed  the  merid 
'onal  parts  corresponding  to  any  latitude  less  than  80°.     Thus 
the  meridional  parts  for  latitude  12°  23'  are    748.9 ; 
"  "  "  57°  42'  are  4260.5. 

TABLE  OF  CORRECTIONS  TO  MIDDLE  LATITUDE,  p.  149. 

This  table  is  used  in  Navigation  for  correcting  the  middle  latitude. 
The  given  middle  latitude  is  to  be  found  either  in  the  first  or  last  verti- 
cal column,  opposite  to  which,  and  under  the  given  difference  of  latitude, 
K;  inserted  the  proper  correction  in  minutes,  to  be  added  to  the  middle 
latitude  to  obtain  the  latitude  in  which  the  meridian  distance  is  accu- 
rately equal  to  the  departure.  Thus,  if  the  middle  latitude  is  41°,  and 
the  difference  of  latitude  14°,  the  correction  will  be  found  to  be  25', 
which,  ndded  to  the  middle  latitude,  gives  the  corrected  middle  latitude 
41°  23' 


A  TABLE 

Of 

LOGARITHMS  OF  NUMBERS 

FROM    1    TO    10,000. 

N. 

Log. 

N. 

Log. 

N. 

Log. 

N. 

Log. 

i 

2 

3 
4 
5 

o.oooooo 
o.3oio3o 
0.477121 
0.602060 
0.698970 

26 

27 
28 

o9 

3o 

.4i4973 
.43i364 
.447i58 
.462398 
.477121 

5i 

52 

53 
54 
55 

i.  7o757o 
i  .716003 
i  .724276 
1.732394 
i.74o363 

76 

77 
78 

79 
80 

.880814 
.886491 
.892096 
.897627 
.903090 

6 

7 
8 

9 

10 

0.778151 
0.845098 
0.903090 
0.954243 

I.  000000 

3i 

32 

33 
34 
35 

.491362 
.5o5i5o 
.5i85i4 
.53x479 
.544o68 

56 

57 
58 
59 
60 

1.748188 
1.755876 
1.768428 
i  .770852 
i.  778161 

8! 

82 
83 
84 
85 

.  908486 
.913814 
.919078 
.924279 
.929419 

ii 

12 

i3 

1.4 

i5 

I.  041893 

I  .079181 

1.113943 

J.  I46I28 

1.176091 

36 
37 
38 

39 
4o 

.5563o3 
.568202 
.579784 
.5qio65 
.602060 

61 

62 
63 
64 
65 

1.786330 
i  .792392 
1.799341 
i.  806180 
i  .812913 

86 
87 
88 
89 
90 

.934498 
.939619 
.944483 
.949390 
.964243 

16 

17 

18 

'9 

20 

I.204I20 

i.23o449 
1.255273 
i  .278754 
i.3oio3o 

4i 

42 

43 
44 
45 

.612784 
.623249 
.633468 
.643453 
.653213 

66 
67 
68 
69 
70 

1.819644 
i  .826076 
1.832609 
i.838849 
1.846098 

91 
92 
93 

9i 
96 

.969041 
.963788 
.968483 
.973128 
.977724 

21 
22 
23 

24 

r 

1.322219 
1.342423 
1.361728 
i.38o2ii 
1.397940 

46 

47 
48 

49 
5o 

.662758 
.672098 
.681241 
.690196 
.698970 

7i 

72 

7/ 

74 

75 

1.861268 
1.867332 
1.863323 
1.869232 
1.876061 

96 

97 
98 

99 

IOO 

.982271 
.986772 
.991226 
995635 

2.OOOOOO 

N.B.  In  the  following  table,  the  two  leading  figures  in  the  first  column 

of  logarithms  are  to  be  prefixed  to  all  the  numbers  of  the  same  horizontal 

line  in  the  next  nine  columns  ;  but  when  a  point  (.)  occurs,  its  place  is  to 

be  supplied  by  a  cipher,  and  the  two  leading  figures  are  to  be  taken  from 

the  next  lower  line. 

LOGARITHMS    OF    .N  u  M  B  E  K  s. 


~N~]   0  j  1 

2 

3 

4  ||  5 

6 

7    8 

9   D. 

100 

oooooo 

o434 

0868 

i3oi 

i?34 

2166 

2598 

3029 

346i 

389i 

43a 

101 

432i 

475i 

SiSi 

5609 

6o38 

6466 

6894 

732I 

7748 

8i74 

428 

102 

8600 

9026 

945i 

9876 

.3oo 

•724 

1147 

i57o 

I993 

24i5 

424 

io3 

oi2837 

3259 

368o 

4ioo 

452i 

494o 

53Co 

5779 

6197 

6616 

419 

io4 

7o33 

745i 

7868 

8284 

8700 

9ii6 

9532 

9947 

.36i 

.775 

4x6 

io5 

02Il89 

i6o3 

2016 

2428 

2841 

3252 

3664 

4o75 

4486 

4896 

412 

1  06 

53o6 

57i5 

6i25 

6533 

6942 

735o 

7757 

8i64 

857i 

8978 

4o8 

X07 

9384 

9789 

.i95 

.600 

ioo4 

i4o8 

1812 

2216 

2619 

3021 

4o4 

1  08 

o33424 

3826 

4227 

4628 

5029 

543o 

583o 

623o 

6629 

•7028 

4oo 

io9 

7426 

7825 

8223 

8620 

9017 

94i4 

9811 

.207 

.602 

.998 

396 

no 

o4i393 

1787 

2182 

2576 

2969 

3362 

3755 

4i48 

454o 

4932 

393 

in 

5323 

57i4 

6io5 

6495 

6885 

7275 

7664 

8o53 

8442 

883o 

389 

112 

9218 

9606 

9993 

.38o 

.766 

n53 

i538 

1924 

2309 

2694 

386 

n3 

o53o78 

3463 

3846 

4230 

46i3 

4996 

5378 

576o 

6142 

6524 

382 

n4 

6905 

7286 

7666 

8o46 

8426 

88o5 

9i85 

9563 

9942 

.320 

379 

ii5 

060698 

io75 

i452 

1829 

2206 

c582  2958 

3333 

37o9 

4o83 

376 

116 

4458 

4832 

5206 

558o 

5953 

6326 

6699 

7o7i 

7443 

78i5 

373 

ii7 

8186 

8557 

8928 

9298 

9668 

..38 

•  4<>7 

.776 

ii45 

i5i4 

369 

ix8 

071882 

225o 

2617 

2985 

3352 

37i8 

4o85 

445i 

48i6 

5i82 

366 

1x9 

5547 

59I2 

6276 

664o 

7004 

7368 

773i 

8094 

8457 

8819 

363 

I2O 

9181 

9543 

9904 

.266 

.626 

.987 

i347 

1707 

206-7 

2426 

36o 

121 

082785 

3i44 

35o3 

386i 

4219 

4576 

4934 

5291 

5647 

6oo4 

357 

N. 

0   |  1 

2 

3 

4   ||  5 

6 

7 

8  |  9 

D. 

434 

43 

87 

i3o 

1  74 

2I7 

260 

3o4 

347 

3ox 

432 

43 

86 

i3o 

i73 

216 

259 

302 

346 

389 

43o 

43 

86 

129 

I72 

2l5 

258 

3oi 

344 

387 

428 

43 

86 

128 

I7I 

214 

257 

3oo 

342 

385 

426 

43 

85 

128 

I70 

213 

256 

298 

34i 

383 

424 

42 

85 

127 

I7O 

212 

254 

297 

339 

382 

422 

42 

84 

127 

169 

211 

253 

295 

338 

38o 

420 

42 

84 

126 

1  68 

210 

252 

294 

336 

378 

4x8 

42 

84 

125 

167 

209 

25l 

293 

334 

376 

4i6 

42 

83 

125 

1  66 

208 

250 

291 

333 

374 

4x4 

4i 

S3 

124 

166 

2O7 

248 

290 

33i 

373 

4l2 

4i 

8-2 

124 

i65 

206 

247 

288 

33o 

371 

4io 

4i 

82 

123 

1  64 

2O5 

246 

287 

328 

369 

4o8 

4i 

82 

122 

i63 

2O4 

245 

286 

326 

367 

4o6 

4i 

81 

122 

162 

203 

244 

284 

325 

365 

4o4  a 

4o 

81 

121 

162 

202 

242 

283 

323 

364 

402  e3 

4o 

80 

121 

161 

201 

a4i 

281 

322 

362 

g 

4oo  ^ 

4o 

80 

120 

1  60 

2OO 

24O 

280 

320 

36o 

a 

398  g 

4o 

80 

II9 

160 

199 

239 

279 

3i8 

358 

CD 

396  .2  < 

4o 

79 

II9 

iSQ 

198 

238 

277 

8x7 

356 

| 

394  g. 

39 

79 

1X8 

1  58 

I97 

236 

276 

3i5 

355 

" 

392  8 

39 

78 

118 

i57 

196 

235 

274 

3x4 

353 

39o  &H 

39 

78 

117 

1  56 

i95 

234 

273 

3l2 

35i 

388 

39 

78 

116 

i55 

i94 

233 

272 

3io 

349 

386 

39 

77 

116 

i54 

i93 

232 

270 

309 

347 

384 

38 

77 

ii5 

1  54 

I92 

230 

269 

3o7 

346 

382 

38 

76 

ii5 

i53 

i9i 

229 

•267 

3o6 

344 

38o 

38 

76 

«4 

l52 

I9O 

228 

266 

3o4 

342 

378 

38 

76 

Ii3 

i5i 

i89 

227 

265 

302 

34o 

376 

38 

75 

Ii3 

i5o 

188 

226 

263 

3oi 

338 

374 

37 

75 

112 

i5o 

187 

224 

262 

299 

337 

372 

3? 

74 

112 

i49 

1  86 

223 

260 

298 

335 

37o 

37 

74 

III 

i48 

i85 

222 

259 

296 

333 

368 

37 

74 

110 

i47 

1  84 

221 

258 

294 

33i 

366    1  37 

73 

110 

1  46 

i83 

220 

256 

293 

329 

364     36 

73 

I09 

146 

182 

218 

255 

29I 

3a8 

362     36 

72 

I09 

i45 

181 

2I7 

253 

29O 

826 

36o    i  36 

72 

1  08 

1  44 

180 

216 

352 

288 

324 

LOGARITHMS    OF    NUMBERS. 


N. 

0 

1 

2 

3 

4     5 

6 

7 

8  J  9  ID.] 

I2S 

oS636o 

6716 

7071 

7426 

778i 

8i36 

8490 

8845 

9198 

9662 

355 

123 

99o5 

.268 

.611 

.963 

i3i5 

1667 

2018 

2370 

2721 

3071 

35i 

124 

093422 

3772 

4l22 

447i 

4820 

6169 

55i8 

5866 

6215 

6662 

349 

125 

6910 

7267 

7604 

796! 

8298 

8644 

8990 

9335 

9681 

..26 

346 

126 

100371 

0716 

1069 

i4o3 

i747 

2091 

2434 

2777 

3119 

3462 

343 

127 

38o4 

4i46 

4487 

4828 

6169 

55io 

585i 

6191 

65.?  i 

6871 

3/i  i 

128 

7210 

7649 

7888 

8227 

8565 

8903 

9241 

9679 

9916 

.253 

338 

129 

110690 

0926 

1263 

1699 

1934 

2270 

2606 

2940 

3275 

3609 

335" 

i3o 

3943 

4277 

46n 

4944 

6278 

56n 

5943 

6276 

6608 

6940 

333 

i3i 

7271 

7603 

7934 

8266 

8595 

8926 

9266 

9686 

99i5 

.245 

33o 

132 

130674 

o9o3 

I23l 

1660 

1888 

2216 

2644 

2871 

3198 

3525 

328 

i33 

3852 

4178 

45o4 

483o 

5x56 

548  1 

58o6 

6i3i 

6456 

6781 

325 

1  34 

7106 

742  9 

7753 

8076 

8399 

8722 

9045 

9368 

9690 

.  .12 

323 

:35 

i3o334 

o655 

o977 

1298 

i6i9 

I939 

2260 

2680 

2900 

32I9 

321 

1  36 

3539 

3858 

4i77 

4496 

48i4 

5i33 

545  1 

6769 

6086 

64o3 

3i8 

i37 

6721 

7o37 

7354 

767i 

7987 

83o3 

8618 

8934 

9249 

9564 

3i5 

1  38 

9879 

.194 

.5o8 

.822 

n36 

i45o 

i763 

2076 

2389 

2702 

3i4 

i39 

i43oi5 

3327 

3639 

3961 

4263 

4574 

4885 

6196 

6607 

6818 

3n 

i4o 

6128 

6438 

6748 

7o58 

7367 

7676 

7985 

8294 

86o3 

8911 

309 

Ui 

9219 

9627 

9835 

.142 

•  449 

.756 

io63 

1370 

i676 

1982 

3o7 

i4a 

162288 

2694 

29OO 

32o5 

35io 

38i5 

4l2O 

4424 

4728 

5o32 

3o5 

i43 

5336 

564o 

5943 

6246 

6549 

6862 

7i54 

7457 

7759 

8061 

3o3 

i44 

8362 

8664 

8965 

9266 

9567 

9868 

.168 

•  469 

.-769 

1068 

3oi 

i45 

i6i368 

1667 

I967 

2266 

2664 

2863 

3i6i 

346o 

3758 

4o55 

299 

i46 

4353 

465o 

4947 

6244 

554i 

5838 

6i34 

643o 

6726 

7022 

297 

i47 

73i7 

7613 

79o8 

8ao3 

8497 

8792 

9086 

938o 

9674 

9968 

296 

i48 

170262 

o555 

o848 

n4i 

i434 

1726 

2019 

23ll 

26o3 

2896 

293 

N.   0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

D. 

358 

'  36 

72 

107 

i43 

i79 

ai5 

261 

286 

322 

356 

36 

7* 

I07 

i4a 

178 

214 

249 

286 

320 

354 

35 

71 

1  06 

142 

I77 

212 

248 

283 

3ig 

352 

35 

70 

1  06 

i4i 

i76 

211 

246 

282 

3i7 

35o    {  35 

7o 

io5 

i4o 

i75 

2IO 

245 

280 

3i5 

348 

35 

7o 

io4 

i39 

i74 

209 

244 

278 

3i3 

346 

35 

69 

io4 

1  38 

i73 

208 

242 

277 

3n 

344 

34 

69 

io3 

1  38 

I72 

206 

241 

275 

3io 

342 

34 

68 

io3 

i37 

I7I 

206 

239 

274 

3o8 

34o 

34 

68 

102 

i36 

I70 

2O4 

238 

272 

3o6 

338 

34 

68 

101 

1  35 

I69 

203 

237 

270 

3o4 

336 

34 

67 

101 

1  34 

168 

2O2 

235 

269 

302 

334 

33 

67 

IOO 

1  34 

i67 

2OO 

234 

267 

3oi 

332  g 

33 

66 

IOO 

i33 

166 

I99 

232 

266 

299 

• 

33o  « 

33 

66 

99 

132 

i65 

I98 

23l 

264 

297 

X 

328  5 

33 

66 

98 

i3i 

1  64 

197 

230 

262 

296 

f  • 

3a6  c  , 

o 

33 

65 

98 

i3o 

i63 

196 

228 

261 

293 

£ 

324   £ 

32 

65 

97 

i3o 

162 

194 

227 

269 

292 

S 

322   g. 

32 

64 

97 

129 

161 

193 

225 

268 

290 

320   2 

32 

64 

96 

128 

1  60 

192 

224 

266 

288 

3i8  ^ 

32 

64 

95 

127 

169 

191 

223 

264 

286 

3i6 

32 

63 

95 

126 

168 

190 

221 

253 

284 

3i4 

3i 

63 

94 

126 

167 

188 

22O 

261 

283 

3l2 

3i 

62 

94 

126 

166 

i87 

218 

260 

281 

3io 

3i 

62 

93 

124 

155 

186 

2I7 

248 

279 

3o8 

3i 

62 

92 

123 

1  54 

i85 

216 

246 

277 

3o6 

3i 

61 

92 

122 

i53 

1  84 

214 

245 

276 

3o4 

3o 

61 

91 

122 

162 

182 

2l3 

243 

274 

302 

3o 

60 

91 

121 

i5i 

181 

211 

242 

272 

3oo 

3o 

60 

9° 

I2O 

1  56 

180 

210 

240 

270 

298 

3o 

60 

89 

119 

149 

i  -79 

209 

238 

268 

296 

3o 

59 

89 

118 

i48 

178 

2O7 

237 

266 

294    i  29 

59 

88 

118 

i47 

176 

206 

235 

265 

LOGARITHMS    OF    NUMBER:*. 


N. 

0 

1 

2 

3 

4 

5 

6 

7 

8  |  9 

D. 

1  4y 

173186 

3478 

3769 

4o6o 

435i 

-,  464  1 

4932 

5222 

55i2  58o2 

291 

160 

6091 

638i 

6670 

6959 

7248 

7536 

7825 

8ii3 

84oi 

8689 

289 

i5i 

8977 

9264 

9552 

9839 

.126 

.4i3 

.699 

.986 

I272 

i558 

287 

1  5?. 

181844 

2129 

24i5 

2700 

2985 

3370 

3555 

3839 

4i23 

44o7 

285 

i53 

469i 

4975 

5259 

5542 

5825 

6108 

639i 

6674 

6956 

7239 

283 

1  54 

7621 

78o3 

8o84 

8366 

8647 

8928 

9209 

949o 

977i 

..5i 

281 

i55 

ioo332 

0612 

0892 

1171 

i45i 

1730 

2OIO 

2289 

2567 

2846 

279 

1  56 

3i25 

34o3 

368i 

3959 

4237 

45i4 

4792 

5o69 

5346 

5623 

278 

1*7 

6900 

6176 

6453 

6729 

7000 

7281 

7556 

7832 

8io7 

8382 

276 

168 

8667 

8932 

9206 

948i 

9755 

..29 

.3o3 

.577 

.85o 

1124 

274 

i59 

201897 

1670 

1943 

2216 

2488 

2761 

3o33 

33o5 

3577 

3848 

272 

;6o 

£120 

439i 

4663 

4934 

5204 

5475 

5746 

60:6 

6286 

6556 

271 

161    6826 

7096 

7365 

7634 

79°4 

8i73 

844  1 

87io 

8979 

9247 

269 

162 

95i5 

9783 

..5i 

.319 

.586 

.853 

I  121 

i388 

i654 

1921 

267 

i63 

212188 

2454 

2720 

2986 

3252 

35i8 

3783 

4o49 

43i4 

4579 

266 

1  64 

4844 

5  1  09 

5373 

5638 

6902 

6166 

643o 

6694 

6957 

722I 

264 

i65 

7484 

7747 

8010 

8273 

8536 

8798 

9060 

9323 

9585 

9846 

262 

166 

220108 

0370 

o63i 

0892 

n53 

i4i4 

i675 

ig36 

2I96 

2456 

261 

167 

2716 

2976 

3236 

3496 

3755 

4oi5 

4274 

4533 

4792 

5o5i 

269 

168 

53o9 

5568 

5826 

6o84 

6342 

6600 

6858 

7u5 

7372 

763o 

258 

169 

7887 

8i44 

84oo 

8657 

89i3 

9I7° 

9426 

9682 

9938 

.i93 

256 

170 

230449 

0704 

0960 

I2l5 

1470 

1724 

i979 

2234 

2488 

2742 

254 

171 

2996 

325o 

35o4 

3757 

4on 

4264 

45i7 

477o 

5o23 

5276 

253 

172 

5528 

578i 

6o33 

6285 

6537 

6789 

7o4i 

7292 

7544 

7795 

262 

i73 

8o46 

8297 

8548 

8799 

9o49 

9299 

955o 

9800 

..5o 

.3oo 

260 

174 

240649 

0799 

io48 

1297 

1  546 

i795 

2044 

2293 

254i 

2790 

249 

i75 

3o38 

3286 

3534 

3782 

4o3o 

4277 

4525 

4772 

5019 

5266 

248 

176 

55i3 

5769 

6006 

6262 

6499 

6745 

699i 

7237 

7482 

7728 

246 

177 

7973 

8219 

8464 

8709 

8954 

9198 

9443 

9687 

9932 

.176 

f  f 

178 

260420 

o664 

0908 

n5i 

i395 

i638 

1881 

2125 

2368 

2?  X> 

243 

179 

2853 

3096 

3338 

358o 

3822 

4o64 

43o6 

4548 

479o 

5o3i 

242 

180 

5273 

55i4 

5755 

5996 

6237 

6477 

6718 

6958 

7i98 

7439 

24  1 

181 

7679 

7918 

8:58 

8398 

8637 

8877 

9116 

9355 

9^94 

9833 

239 

N. 

0 

1 

2 

3 

4 

5 

6 

7  |  8 

0 

D. 

292   f  29 

58 

88 

117 

i46 

i75 

204 

234 

263 

290 

29 

58 

87 

116 

i45 

i74 

203 

232 

261 

288 

29 

58 

86 

IID 

i44 

i73 

202 

230 

269 

286     29 

57 

86 

n4 

i43 

172 

2OO 

229 

2D7 

284     28 

57 

85 

n4 

142 

170 

I99 

227 

256 

282     28 

56 

85 

n3 

i4i 

169 

197 

226 

254 

280 

28 

56 

84 

112 

i4o 

168 

196 

224 

252 

278 

28 

56 

83 

III 

i39 

167 

i95 

223 

25o 

276 

28 

55 

83 

no 

i38 

1  66 

i93 

221 

248 

2?4  j: 

27 

55 

82 

no 

i37 

i64 

I92 

2I9 

247 

272  -g 

27 

54 

82 

io9 

i36 

i63 

I9O 

218 

245 

i 

270  ^ 

27 

54 

81 

1  08 

i35 

162 

I89 

216 

243 

§ 

268  - 

27 

54 

80 

107 

1  34 

161 

188 

2l4 

241 

l< 

266  |. 

27 

53 

80 

106 

i33 

1  60 

186 

213 

239 

I 

264  £ 

26 

53 

79 

1  06 

i3a 

i58 

i85 

211 

238 

3 

262  o^ 

26 

52 

79 

io5 

i3i 

i57 

i83 

2IO 

236 

260  I 

26 

52 

78 

io4 

i3o 

i56 

182 

208 

234 

258  Pk 

26 

52 

77 

io3 

129 

i55 

181 

206 

232 

256 

26 

5i 

77 

IO2 

128 

1  54 

i79 

2O5 

230 

254 

25 

51 

76 

IO2 

127 

152 

i78 

203 

229 

262 

25 

5o 

76 

101 

126 

i5i 

i76 

2O2 

227 

25o 

25 

5o 

75 

IOO 

125 

i5o 

i75 

2OO 

225 

248 

25 

5o 

74 

99 

124 

149 

i74 

198 

223 

246 

25 

49 

74 

98 

123 

i48 

I72 

197 

221 

a44 

24 

49 

73 

98 

122 

i46 

I7I 

i95 

22O 

242 

24 

48 

73 

97 

121 

i45 

169 

194 

218 

240 

24 

48 

72 

96 

120 

i44 

1  68 

192 

216 

LOGARITHMS    OF    NUMBERS. 


N. 

0 

i 

2 

3 

4     5 

6 

7 

8 

9 

D. 

182 

260071 

o3io 

o548 

0787 

IO25 

1263 

i5oi 

i739 

1976 

22l4 

238 

i83 

2461 

2688 

2925 

3i62 

3399 

3636 

3873 

4109 

4346 

458a 

2:7 

i84 

48i8 

5o54 

5290 

5525 

576i 

5996 

6232 

6467 

6702 

6937 

235 

i85 

7172 

7406 

7641 

7875 

8110 

8344 

8578 

8812 

9046 

9279 

234 

186 

95i3 

9746 

9980 

.213 

.446 

.679 

.9I2 

n44 

i377 

1609 

233 

187 

271842 

2074 

23o6 

2538 

2770 

3ooi 

3233 

3464 

3696 

3927 

232 

188 

4:58 

4389 

4620 

485o 

5o8i 

53n 

5542 

5772 

6002 

6232 

230 

189 

6462 

6692 

6921 

7i5: 

738o 

•7609 

7838 

8o67 

8296 

8525 

229 

190 

8754 

8982 

9211 

9439 

9667 

9895 

.123 

.35i 

.578 

.806 

228 

191 

28io33 

I2DI 

i488 

i7i5 

1942 

2169 

2396 

2622 

2849 

8076 

227 

192 

33oi 

3527 

3753 

3979 

42o5 

443  1 

4656 

4882 

5107 

5332 

226 

193 

5557 

5782 

6007 

6456 

6681 

69o5 

7i3o 

7354 

7578 

225 

u;4 

7802 

8026 

8249 

8473 

8696 

8920 

9i43 

9366 

9589 

9812 

223 

i95 

290035 

025? 

o48o 

0702 

0925 

u47 

i369 

iSgi 

i8i3 

2o34 

222 

196 

2256 

2478 

2699 

2920 

3i4i 

3363 

3584 

38o4 

4o25 

4246 

221 

197 

4466 

4687 

4907 

5127 

5347 

5567 

5787 

6oo7 

6226 

6446 

22O 

198 

6665 

6884 

7104 

7323 

7542 

7761 

7979 

8198 

84i6 

8635 

2I9 

199 

8853 

9071 

9289 

95°7 

9725 

9943 

.161 

.378 

.595 

.8i3 

218 

200 

3oio3o 

1247 

1681 

1898 

2Il4 

233i 

2764 

2980 

2I7 

201 

3i96 

34i2 

3628 

3844 

4o59 

4275 

449i 

4706 

4921 

5i36 

216 

202 

535i 

5566 

5781 

5996 

6211 

6425 

6639 

6854 

7068 

7282 

2!4 

203 

7496 

7710 

7924 

8i37 

835i 

8564 

8778 

8901 

9204 

9417 

2!3 

204 

963o 

9843 

..56 

.268 

.48i 

.693 

.9o6 

1118 

i33o 

1  542 

212 

205 

3n754 

1966 

2177 

2389 

2600 

2812 

3o23 

3234 

3445 

3656 

211 

206 

3867 

4o78 

4289 

4499 

4710 

4920 

5i3o 

534o 

555i 

5760 

210 

207 

5970 

6180 

639o 

6599 

6809 

7018 

7227 

7436 

7646 

7854 

2O9 

208 

8o63 

8272 

848  1 

8689 

8898 

9106 

9522 

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52ii38 

1269 

i4oo 

i53o 

1661 

I792 

1922 

2o53 

2i83 

23i4 

333 

s444 

2575 

2705 

2835 

2966 

3o96 

3226 

3356 

3486 

36i6 

i3o 

334 

3746 

3876 

4oo6 

4i36 

4266 

4396 

4526 

4656 

4785 

4gi5 

335 

5o45 

5i74 

53o4 

5434 

5563 

5693 

5822 

595i 

6081 

6210 

129 

336 

6339 

6469 

6598 

6727 

6856 

6985 

7114 

7243 

7372 

75oi 

337" 

763o 

7759 

7888 

8016 

8i45 

8274 

8402 

853i 

8660 

8788 

338 

8917 

9o45 

9174 

9302 

943o 

9559 

9687 

98i5 

9943 

..72 

128 

339 

53o2oo 

o328 

o456 

o584 

07I2 

o84o 

0968 

io96 

1223 

i35i 

34o 

1479 

1607 

1734 

1862 

1990 

2117 

2245 

2372 

25oo 

2627 

34i 

2754 

2882 

3009 

3i36 

3264 

3391 

35i8 

3645 

3772 

3899 

127 

342 

4026 

4i53 

4280 

44o7 

4534 

466i 

4787 

49i4 

5o4i 

5i67 

343 

5294 

542i 

5547 

5674 

58oo 

5927 

6o53 

6180 

63o6 

6432 

126 

344 

6558 

6685 

6811 

6937 

7o63 

7189 

73i5 

744i 

7567 

7693 

345 

7819 

7945 

8071 

8197 

832s 

8448 

8574 

8699 

8825 

8951 

346 

9076 

9202 

9327 

9452 

9578 

97°3 

9829 

9954 

••79 

.204 

125 

347 

5^0339 

o455 

o58o 

0705 

o83o 

o955 

1080 

I2O5 

i33o 

i454 

348 

i579 

1704 

1829 

i953 

2078 

2203 

2327 

2452 

2576 

2701 

349 

2825 

2950 

3o74 

3i99 

3323 

3447 

357i 

3696 

3820 

3o44 

124 

35o 

4o68 

4192 

43i6 

444o 

4564 

4688 

4812 

4936 

5o6o 

5i83 

35i 

53o7 

543  1 

5555 

5678 

58o2 

5925 

6049 

6172 

6296 

6419 

352 

6543 

6666 

6789 

69i3 

7o36 

7i59 

7282 

74o5 

75s9 

7652 

123 

353 

7775 

7898 

8021 

8i44 

8267 

8389 

85i2 

8635 

8758 

8881 

354 

9003 

9126 

9249 

937i 

9494 

96i6 

9739 

9861 

9984 

.106 

355 

55o228 

o35i 

o473 

o595 

o7:7 

o84o 

0962 

1084 

1206 

1328 

122 

356 

i45o 

1572 

i694 

1816 

I938 

2060 

2181 

23o3 

2425 

2547 

357 

2668 

2790 

29II 

3o33 

3i55 

3276 

3398 

35i9 

364o 

3762 

121 

358 

3883 

4oo4 

4l26 

4247 

4368 

4489 

46io 

473i 

4852 

4973 

359 

5094 

52i5 

5336 

5457 

5578 

5699 

5820 

5940 

6061 

6182 

36o 

63o3 

6423 

6544 

6664 

6785 

6905 

7026 

7i46 

7267 

7387 

120 

36  1 

75°7 

7627 

7748 

7868 

7988 

8108 

8228 

8349 

8469 

8589 

N. 

0 

1 

2 

3 

4   ||   5 

6  |  7 

8 

9 

D. 

1  38 

i4 

28 

4i 

55 

69 

83 

97 

no 

124 

i36  « 

i4 

27 

4i 

54 

68 

82 

96 

109 

122 

1  34  fe 

i3 

27 

4o 

54 

67 

80 

94 

107 

121 

f 

132  fit 

i3 

26 

4o 

53 

66 

79 

92 

1  06 

119 

03  , 

i3o  -3 

a  , 

i3 

26 

39 

52 

65 

78 

91 

io4 

117 

1 

128  -2 

i3 

26 

38 

5i 

64 

77 

9° 

IO2 

ii5 

Qi 

126  | 

i3 

25 

38 

5o 

63 

76 

88 

101 

ii3 

124   0 

12 

25 

37 

5o 

62 

74 

87 

99 

112 

j  122  & 

12 

24 

37 

40 

61 

73 

85 

98 

110 

(iao 

12 

24   36   48 

60 

72 

84 

96 

108 

UOGARITHMS     OF     NUMBERS. 


N. 

0 

i 

2 

3 

4 

5 

6 

7 

8 

9 

D.  1 

"  362 

5587o9 

8829 

8948 

9068 

9i88 

93o8 

9428 

9548 

9667 

9787 

120 

363 

99°7 

..26 

.i46 

.265 

.385 

.5o4 

.624 

.743 

.863 

.982 

364 

56noi 

1221 

1  34o 

i459 

i578 

1698 

1817 

I936 

2o55 

2174 

119 

365 

2293 

24l2 

253i 

265o 

2769 

2887 

3oo6 

3i25 

3244 

3362 

366 

348  1 

36oo 

37i8 

3837 

3955 

4074 

4192 

43ii 

4429 

4548 

5<57 

4666 

4784 

4903 

5021 

5i39 

5257 

5376 

5494 

56i2 

573o 

118 

368 

5848 

5966 

6084 

6202 

6320 

6437 

6555 

6673 

679i 

69o9 

369 

•7026 

7i44 

7262 

7379 

7497 

7614 

7732 

7849 

7967 

8o84 

37o 

8202 

83i9 

8436 

8554 

8671 

8788 

8905 

9023 

9i4o 

9257 

117 

37i 

9374 

949i 

9608 

9725 

9842 

9959 

..76 

.193 

.3o9 

.426 

372 

57o543 

0660 

0776 

0893 

IOIO 

1126 

1243 

i359 

1476 

I092 

373 

I7o9 

i825 

1942 

2o58 

2174 

2291 

2407 

2523 

2639 

2755 

116 

374 

2872 

2988 

3io4 

322O 

3336 

3452 

3568 

3684 

38oo 

39i5 

375 

4o3i 

4i47 

4263 

4379 

4494 

46  10 

4726 

484i 

4957 

5072 

376 

5i88 

53o3 

54i9 

5534 

565o 

5765 

588o 

5996 

6111 

6226 

n5 

377 

634i 

6457 

6572 

6687 

6802 

6917 

7o32 

7i47 

7262 

7377 

378 

7492 

76o7 

7722 

7836 

795i 

8066 

8181 

8295 

84io 

8525 

379 

8639 

8754 

8868 

8983 

9°97 

O2I2 

9326 

944i 

9555 

9669 

n4 

38o 

9784 

9898 

.  .  12 

.126 

.241 

.355 

.469 

.583 

.697 

.811 

38i 

58o925 

io39 

ii53 

1267 

i38i 

i495 

1608 

I722 

i836 

i95o 

382 

2o63 

2I77 

2291 

2404 

25i8 

263i 

2745 

2858 

2972 

3o85 

383 

3i99 

33i2 

3426 

3539 

3652 

3765 

3879 

3992 

4io5 

4218 

ii3 

384 

433i 

4444 

4557 

4670 

4783 

4896 

5009 

5l22 

5235 

5348 

385 

546i 

5574 

5686 

5799 

59I2 

6024 

6i37 

625o 

6362 

6475 

386 

6587 

6700 

6812 

6925 

7o37 

7149 

7262 

7374 

7486 

7599 

112 

387 

77ii 

7823 

7935 

8047 

8160 

8272 

8384 

8496 

8608 

8720 

388 

8832 

8944 

go56 

9167 

9279 

939i 

95o3 

96i5 

9726 

9838 

389 

995o 

..61 

.i73 

.284 

.396 

.507 

.619 

.73o 

.842 

.953 

39o  ,  59io65 

2:76 

1287 

i399 

i5io 

1621 

1732 

i843 

i955 

2066 

III 

39i  !   2i77 

2288 

2399 

25lO 

2621 

2732 

2843 

2954 

3o64 

3i75 

392 

3286 

3397 

35o8 

36i8 

3729 

384o 

3950 

4o6i 

4171 

4282 

393 

4393 

45o3 

46i4 

4724 

4834 

4945 

5o55 

5i65 

5276 

5386 

IIO 

394 

5496 

56o6 

6717 

5827 

5937 

6047 

6i57 

6267 

6377 

6487 

395 

6597 

6707 

6817 

6927 

7o37 

7146 

7256 

7366 

7476 

7586 

396 

7695 

78o5 

7914 

8024 

8i34 

8243 

8353 

8462 

8572 

8681 

397 

879i 

89oo 

9009 

9II9 

9228 

9337 

9446 

9556 

9665 

9774 

109 

398 

9883 

9992 

.101 

.210 

.3i9 

.428 

.537 

.646 

.755 

.864 

399 

6oo973 

1082 

1191 

I299 

i4c8 

i5i7 

1625 

i734 

i843 

i95i 

4oo 

2060 

2l69 

2277 

2386 

2494 

26o3 

27II 

28l9 

2928 

3o36 

1  08 

4oi 

3i44 

3253 

336i 

3469 

3577 

3686 

3794 

39O2 

4oio 

4n8 

402 

4226 

4334 

4442 

455o 

4658 

4766 

4874 

4982 

5o89 

5i97 

4o3 

53o5 

54  1  3 

552i 

5628 

5736 

5844 

595i 

6o59 

6166 

6274 

4o4 

essi 

6489 

65g6 

6704 

6811 

6919 

•7026 

7i33 

7241 

7348 

I07 

4o5 

7455 

7562 

7669 

7777 

7884 

7991 

8098 

82o5 

83i2 

84i9 

4o6 

8526 

8633 

8740 

8847 

8954 

9061 

9l67 

9274 

938i 

9488 

4o7 

9594 

9701 

9808 

99i4 

.  .21 

.128 

.234 

.34i 

.447 

.554 

4o8 

610660 

0767 

o873 

°979 

1086 

1192 

1298 

i4o5 

i5ii 

161-7 

1  06 

N. 

0   |  1 

2 

3 

4  ||   5 

6 

7 

8 

9 

D. 

IIO 

12 

24 

36 

48 

60 

71 

83 

95 

I07 

118 

12 

24 

35 

47 

59 

71 

83 

94 

1  06 

117 

12 

23 

35 

47 

59 

7° 

82 

94 

io5 

zi6  £ 

12 

23 

35 

46 

58 

7° 

81 

93 

104 

n5  « 

12 

23 

35 

46 

58 

69 

81 

92 

104 

8 

n4  - 

II 

23 

34 

46 

5? 

68 

80 

91 

io3 

1 

n3  | 

II 

23 

34 

45 

57 

68 

79 

9° 

102 

*c 

112  £ 

II 

22 

34 

45 

56 

67 

78 

9° 

101 

p 

III   Q. 

II 

22 

33 

44 

56 

67 

78 

89 

IOO 

110  2 

II 

22 

33 

44 

55 

66 

77 

88 

99 

109  ^ 

II 

22 

33 

44 

55 

65 

76 

87 

98 

1  08 

II 

22 

32 

43 

54 

65 

76 

86 

97 

107   i  ii 

21 

32   43 

54 

64 

75 

86 

96 

10 


LOGARITHMS    OP    NUMBERS. 


N. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9   D. 

409 

611723 

l829 

i936 

2O42 

2148 

2254 

236o 

2466 

2572 

2678 

1  06 

4io 

2784 

289o 

2996 

3l02 

3207 

33i3 

34i9 

3525 

363o 

3736 

4n 

3842 

3947 

4o53 

4i5q 

4264 

437o 

4475 

458i 

4686 

4-792 

412 

4897 

5oo3 

5io8 

52i3 

53i9 

5424 

5529 

5634 

574o 

5845 

io5 

4i3 

595o 

6o55 

6160 

6265 

6370 

6476 

658i 

6686 

679o 

6895 

4i4 

7000 

7io5 

7210 

73i5 

7420 

7525 

7629 

7734 

7839 

7943 

4i5 

8o48 

8i53 

8257 

8362 

8466 

857i 

8676 

8780 

8884 

8989 

4i6 

9o93 

9i98 

93o2 

94o6 

95n 

96i5 

9719 

0824 

9928 

.Tall 

104 

4ry 

62oi36 

O24O 

o344 

o448 

o552 

o656 

0760 

o864 

0968 

1072 

4i8 

1176 

1280 

1  384 

i488 

1592 

i695 

1799 

I9o3 

2OO7 

2IIO 

419 

22l4 

23i8 

2421 

2525 

2628 

273,2 

2835 

2939 

3o42 

3i46 

420 

3249 

3353 

3456 

3559 

3663 

3766 

3869 

3973 

4o76 

4179 

io3 

4-21 

4282 

4385 

4488 

459i 

4695 

4798 

4901 

5oo4 

5io7 

5210 

422 

53i2 

54i5 

55i8 

562i 

5724 

5827 

5929 

6o32 

6i35 

6238 

428 

634o 

6443 

6546 

6648 

6751 

6853 

6956 

7o58 

7161 

7263 

424 

7366 

7468 

7571 

7673 

7775 

7878 

7980 

8082 

8i85 

8c87 

102 

425 

8389 

849i 

8593 

8695 

8797 

8900 

9002 

9io4 

9206 

93o8 

426 

94io 

95l2 

96i3 

97i5 

9817 

9919 

.  .21 

.123 

.224 

.326 

427 

63o428 

o53o 

o63i 

o733 

o835 

o936 

io38 

n39 

1241 

1  342 

428 

i444 

i545 

1647 

1748 

1849 

ig5i 

2O52 

2i53 

2255 

2356 

101 

429 

2457 

2559 

2660 

2761 

2862 

2963 

3o64 

3i65 

3266 

3367 

43o 

3468 

3569 

367o 

377i 

3872 

3973 

4074 

4i75 

4276 

4376 

43i 

4477 

4578 

4679 

4779 

488o 

4981 

5o8i 

5i82 

5283 

5383 

IOO 

432 

5484 

5584 

5685 

5785 

5886 

5986 

6087 

6i87 

6287 

6388 

433 

6488 

6588 

6688 

6789 

6889 

6989 

7089 

7i89 

729o 

739° 

434 

749o 

759o 

7600 

779° 

7890 

799° 

8090 

8i9o 

8290 

8389 

435 

8489 

8589 

8689 

8789 

8888 

8988 

9088 

9i88 

9287 

9387 

99 

436   9486 

9586 

9686 

9785 

9885 

9984 

..84 

.i83 

.283 

.382 

437 

64o48i 

o58i 

0680 

0779 

0879 

0978 

1077 

1177 

I276 

i375 

438 

1474 

1573 

1672 

1771 

1871 

i97o 

2069 

2168 

2267 

2366 

439 

2465 

2563 

2662 

2761 

2860 

2959 

3o58 

3i56 

3255 

3354 

44o 

3453 

355i 

365o 

3749 

3847 

3946 

4o44 

4i43 

4242 

434o 

98 

44i 

4439 

4537 

4636 

4734 

4832 

4g3i 

5029 

5127 

5226 

5324 

442 

5422 

552i 

56i9 

5717 

58i5 

59i3 

60  1  1 

6110 

6208 

63o6 

443 

64o4 

65o2 

6600 

6698 

6796 

6894 

6992 

7o89 

7187 

7285 

444 

7383 

748  1 

7579 

7676 

7774 

7872 

7969 

8067 

8i65 

8262 

445 

836o 

8458 

8555 

8653 

875o 

8848 

8945 

9o43 

9i4o 

9237 

97 

446 

9335 

9432 

953o 

9627 

9724 

9821 

9919 

..16 

.n3 

.210 

447 

65o3o8 

o4o5 

o5o2 

0599 

0696 

o793 

0890 

o987 

1084 

1181 

448 

1278 

1375 

1472 

i56g 

1666 

I762 

i859 

i956 

2o53 

2i5o 

449 

2246 

2343 

2440 

2536 

2633 

273o 

2826 

2923 

3019 

3n6 

45o 

32i3 

33o9 

34o5 

35o2 

3598 

3695 

379i 

3888 

3984 

4o8o 

96 

45i 

4177 

4273 

4369 

4465 

4562 

4658 

4754 

485o 

4946 

5o42 

452 

5i38 

5235 

533i 

5427 

5523 

5619 

57i5 

58io 

5906 

6002 

453 

6o98 

6i94 

6290 

6386 

6482 

6577 

6673 

6769 

6864 

6960 

454 

7o56 

7162 

7247 

7343 

7438 

7534 

7629 

7725 

7820 

•7916 

455 

Sou 

8107 

8202 

8298 

8393 

8488 

8584 

8679 

8774 

887o 

95 

456 

8965 

9o6o 

9i55 

925o 

9346 

944i 

9536 

963i 

9726 

9821 

457 

99i6 

.  .11 

.  1  06 

.201 

.296 

.39I 

.486 

.58i 

.676 

.77i 

N. 

0 

I 

2 

3 

4 

5 

6 

7 

8 

9 

1). 

106 

ii 

21 

32 

42 

53 

64 

74 

85 

95 

io5 

ii 

21 

32 

42 

53 

63 

74 

84 

95 

1 

104  g 

10 

21 

3i 

42 

52 

62 

73 

83 

94 

io3  £ 

10 

21 

3i 

4i 

52 

62 

72 

82 

93 

g 

102  _ 

10 

2O 

3i 

4i 

5i 

61 

7i 

82 

92 

©  ^  101   g  , 

*"•  j        O 

10 

20 

3o 

4o 

5i 

61 

71 

81 

91 

,0)          •  —  ' 

fe   100  tj 

IO 

2O 

3o 

4o 

5o 

60 

7° 

80 

9° 

6 

99  c. 

IO 

2O 

3o 

4o 

5o 

59 

69 

79 

So 

S8  I 

IO 

20 

29 

39 

49 

59 

69 

78 

88 

97  * 

IO 

*9 

29 

39 

49 

58 

68 

78 

87 

96 

IO 

'9 

2Q 

38 

48 

58 

67 

77 

86 

LOGARITHMS    OF    IN  UMBERS. 


N.    0 

1 

2 

3 

4  1   5 

6 

7 

8 

9 

-*H 

458 

66o865 

0960 

io55 

ii5o 

1245 

i339 

i434 

l529 

1623 

1718 

9H 

409 

i8i3 

1907 

2OO2 

2O96 

2I9I 

2286 

238o 

2475 

2569 

2663 

46o 

2758 

2852 

2947 

3o4i 

3i35 

323o 

3324 

34i8 

35i2 

3607 

94 

46i 

3701 

3795 

3889 

3983 

4078 

4172 

4266 

436o 

4454 

4548 

462 

4642 

4736 

483o 

4924 

5oi8 

5lI2 

5206 

5299 

5393 

5487 

463 

558i 

5675 

5769 

5862 

5956 

6o5o 

6i43 

6237 

633i 

6424 

464 

65i8 

6612 

6705 

6799 

6892 

6986 

7°79 

7173 

7266 

736o 

465 

7453 

7546 

7640 

7733 

7826 

7920 

8oi3 

8106 

8i99 

8293 

93 

466 

8386 

8479 

8572 

8665 

8759 

8852 

8945 

9o38 

9i3i 

9224 

467 

93i7 

9410 

95o3 

9596 

9689 

0782 

9875 

9967 

..60 

.i53 

468 

670246 

oSSg 

o43i 

OD24 

0617 

0710 

0802 

o895 

o988 

1080 

469 

1173 

1265 

i358 

i45i 

j543 

i636 

1-728 

1821 

I9i3 

2OO5 

4yo 

2098 

2190 

2283 

2375 

2467 

256o 

2652 

2744 

2836 

2929 

92 

4yi 

3021 

3n3 

32o5 

3297 

3390 

3482 

3574 

3666 

3758 

385o 

472 

3942 

4o34 

4126 

4218 

43io 

4402 

4494 

4586 

4677 

4769 

4?3 

486i 

4953 

5o45 

5i37 

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L  O  C,       R  I  T  II  M  3     OF     N  U  M  B  E  K  S. 


N. 

0 

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3 

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7004 

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1024 

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1589 

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2555 

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4079 

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6397 

6476 

6556 

6635 

6715 

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6874 

6954 

7o34 

7.II3 

546 

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7272 

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7749 

7829 

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79 

547 

7987 

8067 

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8225 

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8384 

8463 

8543 

8622 

8701 

548 

8781 

8860 

8939 

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9968 

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1  546 

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553 

2725 

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4449 

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5o75 

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5387 

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562i 

5699 

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69.33 

6011 

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64oi 

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6556 

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6868 

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7023 

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N. 

0 

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-7 

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52 

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62   70 

LOGARITHMS    OF    .NUMBERS. 


13 


N. 

0 

1 

2 

3 

4       5 

6 

7 

8 

9 

D. 

559 

7474i2 

7489 

7567 

7645 

7722 

7800 

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7955 

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78 

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8188 

8266 

8343 

8421 

8498 

857G 

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873i 

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8885 

77 

56i 

8963 

9o4o 

9n8 

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9272 

935o 

9427 

95o4 

9582 

9659 

662 

9736 

98i4 

989i 

9968 

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.123 

.200 

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.354 

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563 

75o5o8 

o586 

o663 

0740 

0817 

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0971 

io48 

I  125 

1  202 

564 

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i356 

i433 

i5io 

i587 

1  664 

1741 

1818 

i895 

I972 

565 

2048 

2125 

22O2 

2279 

2356 

2433 

25o9 

2586 

2663 

2740 

566 

2816 

2893 

2970 

3o47 

3i23 

320O 

3277 

3353 

343o 

35o6 

56y 

3583 

366o 

3736 

38i3 

3889 

3966 

4042 

4u9 

4i95 

4272 

568 

4348 

4425 

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4578 

4654 

4?3o 

4807 

4883 

496o 

5o36 

& 

56g 

5lI2 

5i89 

5265 

534i 

54i7 

5494 

557o 

5646 

5722 

5799 

5yo 

5875 

595i 

6o27 

6io3 

6180 

6256 

6332 

64o8 

6484 

656o 

57i 

6636 

6712 

6788 

6864 

6940 

7016 

700.2 

7i68 

7244 

7320 

572 

7396 

7472 

7548 

7624 

7700 

7775 

785i 

7927 

8oo3 

8079 

573 

8i55 

823o 

83o6 

8382 

8458 

8533 

86o9 

8685 

8761 

8836 

574 

89ia 

8988 

9o63 

9i39 

92l4 

929o 

9366 

944i 

95i7 

9592 

575 

9668 

9743 

9819 

9894 

9970 

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.121 

.  i96 

.272 

.34? 

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576 

760422 

o498 

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o649 

0724 

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1025 

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577 

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i326 

I4O2 

1477 

i552 

1627 

I702 

1778 

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578 

I928 

2003 

2078 

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2228 

23o3 

2378 

2453 

2629 

2604 

579 

2679 

2754 

2829 

29o4 

2978 

3o53 

3i28 

32o3 

3278 

3353 

58o 

3428 

35o3 

3578 

3653 

3727 

38o2 

3877 

3952 

4027 

4ioi 

58i 

4176 

425i 

4326 

44oo 

4475 

455o 

4624 

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4774 

4848 

582 

4923 

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5o72 

5i47 

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537o 

5445 

5520 

5594 

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5743 

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5966 

6o4i 

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6264 

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584 

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6785 

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6933 

7007 

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585 

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7675 

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586 

7898 

7972 

8o46 

8120 

8i94 

8268 

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84i6 

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8564 

587 

8638 

8712 

8786 

8860 

8934 

9oo8 

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9:56 

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93o3 

588 

9377 

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95?5 

9599 

9673 

9746 

9820 

9894 

9968 

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589 

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1661 

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1881 

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2028 

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2175 

2248 

73 

592 

2322 

2395 

2468 

254* 

26i5 

2688 

2762 

2835 

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2081 

593 

3o55 

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3274 

3348 

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3494 

3567 

364o 

3713 

594 

3786 

386o 

3933 

4oo6 

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4225 

4298 

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4663 

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5028 

5ioo 

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596 

5246 

53i9 

5392 

5465 

5538 

56io 

5683 

5756 

5829 

5902 

597 

5974 

6047 

6120 

6i93 

6265 

6338 

64n 

6483 

6556 

6629 

698 

6701 

6774 

6846 

69i9 

6992 

7064 

7i37 

•7209 

7282 

7354 

599 

7427 

7499 

7572 

7644 

77i7 

7789 

7862 

7934 

8006 

8079 

72 

600 

8i5i 

8224 

8296 

8368 

844  1 

85i3 

8585 

8658 

873o 

8802 

601 

8874 

8947 

9OI9 

9o9i 

9i63 

9236 

93o8 

938o 

9452 

9524 

602 

9596 

9669 

974i 

98i3 

9885 

9957 

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.i73 

.245 

6o3 

780317 

o389 

o46i 

o533 

o6o5 

0677 

0749 

0821 

o893 

0965 

6o4 

1037 

no9 

1181 

1253 

i324 

i396 

i468 

i54o 

1612 

1  684 

6o5 

1755 

1827 

i899 

1971 

2042 

2Il4 

2186 

2258 

2329 

2401 

606 

2473 

2544 

2616 

2688 

2759 

283i 

2902 

2974 

3o46 

3n7 

6o7 

3i89 

3260 

3332 

34o3 

3475 

3546 

36i8 

3689 

376i 

3832 

7* 

608 

39o4 

3975 

4o46 

4n8 

4i89 

4261 

4332 

44o3 

4475 

4546 

609 

46i7 

4689 

476o 

483i 

4902 

4974 

5o45 

5n6 

5i87 

5259 

610 

533o 

54oi 

5472 

5543 

56i5 

5686 

5757 

5828 

5899 

5970 

6n 

6o4i 

6112 

6i83 

6254 

6325 

6396 

6467 

6538 

66o9 

6680 

N. 

0   |  1 

2 

3 

4 

5 

6 

7 

8 

9 

D. 

77  » 

8 

i5 

23 

3i 

39 

46 

54 

62 

69 

S 

76  £ 

8 

i5 

23 

3o 

38 

46 

53 

61 

68 

| 

75  £ 

8 

i5 

23 

3o 

38 

45 

53 

60 

68 

8- 

74  C' 

7 

T  * 

1  v> 

23 

3o 

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44 

52 

59 

67 

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73  g. 

7 

i5 

22 

29 

37   44 

5i 

58 

66 

3 

72  2 

7 

i4 

22 

29 

36 

43 

5o 

58 

65 

71  * 

7 

i4 

21 

28 

36 

43 

5o 

5? 

64 

LOGARITHMS    OF    LUMBERS. 


N. 

0 

.1 

2 

3 

4 

5 

6 

7 

8    9  |  D. 

612 

786751 

6822 

6893 

6964 

7035 

7106 

7i77 

7248 

73l(; 

7390 

71 

6i3 

746o 

753i 

7602 

7673 

7744 

7810 

7885 

7956 

8027 

8008 

6i4 

8168 

8239 

83io 

838i 

845  1 

8522 

8593 

8663 

8734  88o4 

6i5 

8875 

8946 

9016 

9087 

9l57 

9228 

9299 

9369 

944o  95io 

616 

958r 

965i 

9722 

9792 

9863 

9933 

...4 

.-74 

.i44  .2i5 

70 

617 

790285 

o356 

0426 

0496 

o567 

o637 

0707 

o778 

o848 

0918 

618 

0988 

loSg 

1129 

1199 

1269 

i34o 

i4io 

i48o 

i55o 

1620 

619 

1691 

1761 

i83i 

1901 

1971 

2041 

2III 

2181 

2252 

2322 

620 

2392 

2462 

2532 

2602 

2672 

2742 

0812 

2882 

2952 

3O22 

621 

3092 

3i62 

323i 

33oi 

3371 

344i 

35u 

358i 

365i 

3721 

622 

3790 

386o 

SgSo 

4ooo 

4070 

4i39 

4209 

4279 

4349 

44i8 

623 

4488 

4558 

4627 

4697 

4767 

4836 

4906 

4976 

5o45 

5n5 

624 

5i85 

5254 

5324 

5393 

5463 

5532 

56o2 

5672 

574i 

58n 

626 

588o 

5949 

6019 

6088 

6i58 

6227 

629-7 

6366 

6436 

65o5 

69 

626 

6574 

6644 

6713 

6782 

6852 

6921 

6990 

7060 

7129 

7198 

627 

7268 

7337 

7406 

7475 

7545 

76l4 

7683 

7752 

7821 

7890 

628 

7960 

8029 

8098 

8167 

8236 

83o5 

8374 

8443 

85i3 

8582 

629 

865i 

8720 

8789 

8858 

8927 

8996 

9065 

9i34 

9203 

9272 

63o 

934i 

9409 

9478 

9547 

9616 

9685 

9754 

9823 

9892 

9961 

63i 

800029 

0098 

0167 

0236 

o3o5 

o373 

0442 

o5u 

o58o 

o648 

63a 

0717 

0786 

o854 

0923 

0992 

1061 

1129 

1198 

1266 

i335 

633 

i4o4 

1472 

i54i 

1609 

1678 

1747 

1816 

1  884 

1952 

2021 

634 

2089 

2i58 

2226 

2295 

2363 

2432 

25oo 

?,568 

2637 

2705 

635 

2774 

2842 

2910 

2979 

3o47 

3n6 

3i84 

3252 

332i 

3389 

68 

636 

3457 

3525 

3594 

3662 

373o 

3798 

3867 

SgSS 

4oo3 

4071 

637 

4i39 

4208 

4276 

4344 

44i2 

448o 

4548 

46i6 

4685 

4753 

638 

4821 

4889 

4957 

5o25 

SogS 

5i6i 

5229 

5297 

5365 

5433 

639 

55oi 

5569 

5637 

57o5 

5773 

584i 

5908 

5976 

6o44 

6112 

64o 

6180 

6248 

63i6 

6384 

645  1 

65i9 

658J 

6655 

6723 

6790 

64  1 

6858 

6926 

6994 

7061 

7129 

7197 

7264 

7332 

74oo 

7467 

642 

7535 

7603 

7670 

7738 

7806 

7873 

794  1 

8008 

8076 

8i43 

643 

8211 

8279 

8346 

84i4 

848  1 

8549 

8616 

8684 

875i 

8818 

C7 

644 

8886 

8953 

9021 

9088 

9i56 

9223 

9290 

9358 

9425 

9492 

645 

9560 

9627 

9694 

9762 

9829 

9896 

9964 

..3i 

..98 

.i65 

646 

810233 

o3oo 

o367 

o434 

o5oi 

0569 

o636 

0703 

0770 

o837 

647 

0904 

0971 

1039 

1106 

n73 

1240 

1307 

i374 

i44.i 

i5o8 

648 

i575 

1642 

1709 

1776 

i843 

1910 

1977 

2044 

2III 

2178 

649 

2245 

23l2 

2379 

2445 

25l2 

2579 

2646 

2713 

2780 

2847 

65o 

2913 

2980 

3o47 

3n4 

3i8i 

3247 

33i4 

338i 

3448 

35i4 

65i 

358i 

3648 

37i4 

3781 

3848 

3914 

398i 

4o48 

4u4 

4i8i 

652 

4248 

43i4 

438i 

444? 

45i4 

458i 

4647 

47i4 

4780 

4847 

653 

4gi3 

4980 

5o46 

5n3 

5i79 

5246 

53i2 

5378 

5445 

55n 

66 

654 

5578 

5644 

57n 

5777 

5843 

Sgio 

5976 

6042 

6109 

6i75 

655 

6241 

63o8 

6374 

644o 

65o6 

6573 

6639 

67o5 

6771 

6838 

656 

6904 

6970 

7o36 

7102 

7169 

7235 

73oi 

7367 

7433 

7499 

657 

7565 

763i 

7698 

7764 

783o 

7896 

7962 

8028 

8094 

8160 

658 

8226 

8292 

8358 

8424 

8490 

8556 

8622 

8688 

8754 

8820 

659 

8885 

895i 

9017 

9083 

9149 

9215 

928l 

9346 

9412 

9478 

660 

9544 

9610 

9676 

974i 

9807 

9873 

9939 

...4 

..70 

.i36 

661 

820201 

0267 

o333 

o399 

o464 

o53o 

o595 

0661 

0727 

0792 

662 

o858 

0924 

0989 

io55 

II2O 

1186 

I25l 

i3i7 

1  382 

i448 

663 

i5i4 

i579 

1  645 

1710 

i775 

i84i 

1906 

I972 

2037 

2103 

65 

664 

2168 

2233 

2299 

2364 

2430 

24g5 

256o 

2626 

2691 

2756 

N. 

0   ]  1 

2 

3 

4 

5 

6 

7 

8 

9 

D. 

7i  « 

7 

i4 

21 

28 

36 

43 

5o 

57 

64 

• 

05 

7°  S 

7 

i4 

21 

28 

35 

42 

49 

56 

63 

W 

a 

69  p^ 

7 

i4 

21 

28 

35 

4i 

48 

55 

62 

S- 

68  ri« 

7 

i4 

2O 

27 

34 

4i 

48 

54 

61 

£ 

e?  a 

7 

i3 

20 

27 

34 

4o 

47 

54 

60 

P 

66  g 

7 

i3 

20 

26 

33 

4o 

46 

53 

59 

65  * 

7 

i3 

20 

26 

33 

39 

46 

52 

59 

LOGARITHMS    OF    NUMBERS. 


15 


N. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

D. 

665 

822822 

2887 

2952 

3oi8 

3o83 

3i48 

32i3 

3279 

3344 

34o9 

65 

666 

3474 

3539 

36o5 

367o 

3735 

38oo 

3865 

3930 

3996 

4o6i 

667 

4126 

4191 

4256 

4321 

4386 

445  1 

45i6 

458i 

4646 

4711 

668 

4776 

484  1 

4906 

4971 

5o36 

5ioi 

5i66 

523i 

5296 

536i 

669 

5426 

5491 

5556 

562i 

5686 

575i 

58i5 

588o 

5945 

6010 

670 

6075 

6i4o 

6204 

6269 

6334 

6399 

6464 

6528 

6593 

6658 

671 

6723 

6787 

6852 

6917 

6981 

7o46 

7ni 

7175 

7240 

73o5 

672 

7369 

7434 

7499 

7563 

•7628 

•7692 

7757 

7821 

7886 

795i 

673 

8oi5 

8o'8o 

8i44 

8209 

8273 

8338 

8402 

8467 

853i 

8595 

64 

674 

8660 

8724 

8789 

8853 

8918 

8982 

9046 

9111 

9i75 

9239 

675 

93o4 

936? 

9432 

9497 

956i 

9625 

9690 

9754 

98i8 

9882 

676 

9947 

..n 

..75 

.i39 

.204 

.268 

.332 

.396 

.46o 

.525 

677 

83o589 

o653 

07I7 

0781 

0845 

0909 

°973 

1037 

1  1  02 

1166 

678 

I23o 

1294 

i358 

1422 

i486 

i55o 

i6i4 

1678 

1742 

1806 

679 

1870 

1934 

1998 

2062 

2126 

2189 

2253 

2317 

238i 

2445 

680 

2509 

2573 

2637 

2700 

2764 

2828 

2892 

2956 

3020 

3o83 

68  1 

3i47 

3211 

3275 

3338 

3402 

3466 

353o 

SSgS 

3657 

3721 

682 

3784 

3848 

3912 

3975 

4o39 

4io3 

4i66 

423o 

4294 

4357 

683 

4421 

4484 

4548 

46n 

4675 

4739 

4802 

4866 

4929 

4993 

684 

5o56 

5l2O 

5i83 

5247 

53io 

5373 

5437 

55oo 

5564 

5627 

63 

685 

569i 

5754 

58i7 

588i 

5944 

6007 

6o7i 

6r34 

6197 

6261 

686 

6324 

6387 

645  1 

65i4 

6577 

664i 

67o4 

6767 

683o 

6894 

687 

6957 

•7020 

7o83 

7i46 

72IO 

7273 

7336 

7399 

7462 

7525 

688 

7588 

7652 

771.5 

7778 

784i 

79o4 

•796-7 

8o3o 

8093 

8i56 

689 

8210 

8282 

8345 

84o8 

847i 

8534 

8597 

8660 

8723 

8786 

690 

8849 

8912 

8975 

9o38 

9101 

9164 

9227 

9289 

9352 

94i5 

691 

9478 

954i 

9604 

9667 

9729 

9792 

9855 

9918 

9981 

..43 

692 

84oio6 

0169 

0232 

0294 

0357 

O42O 

0482 

o545 

0608 

o67i 

693 

o733 

0796 

oSSg 

0921 

0984 

io46 

1109 

1172 

1234 

I297 

694 

i359 

1422 

i485 

i547 

1610 

l672 

i735 

1797 

1860 

1922 

695 

i985 

2047 

2IIO 

2I72 

2235 

229-7 

236o 

2422 

2484 

2547 

62 

696 

2609 

2672 

2734 

2796 

2859 

2921 

2983 

3o46 

3io8 

3i7o 

697 

3233 

3295 

3357 

3420 

3482 

3544 

36o6 

3669 

373i 

3793 

698 

3855 

39i8 

3980 

4042 

4io4 

4i66 

4229 

4291 

4353 

44i5 

699 

4477 

4539 

46oi 

4664 

4726 

4788 

485o 

4912 

4974 

5o36 

700 

5098 

5i6o 

5222 

5284 

5346 

54o8 

547o 

5532 

5594 

5656 

701 

57i8 

578o 

5842 

5go4 

5966 

6028 

6090 

6i5i 

6213 

6275 

702 

6337 

6399 

646  1 

6523 

6585 

6646 

67o8 

6770 

6832 

6894 

7o3 

6955 

7017 

7079 

7i4i 

7202 

•7264 

7326 

7388 

7449 

75n 

704 

7573 

7634 

7696 

7758 

7819 

788i 

7943 

8oo4 

8066 

8128 

7<>5 

8189 

825i 

83i2 

8374 

8435 

8497 

8559 

8620 

8682 

8743 

706 

88o5 

8866 

8928 

8989 

905  1 

9112 

9i74 

9235 

9297 

9358 

61 

707 

9419 

948i 

9542 

9604 

9665 

9726 

9788 

9849 

9911 

9972 

708 

85oo33 

oogS 

oi56 

O2I7 

0279 

o34o 

o4oi 

0462 

o524 

o585 

709 

o646 

0707 

0769 

o83o 

0891 

0952 

1014 

1075 

n36 

1197 

710 

1258 

1320 

i38i 

1442 

i5o3 

1  564 

1625 

1686 

1747 

1809 

711 

i87o 

ig3i 

1992 

2o53 

2Il4 

2I75 

2236 

2297 

2358 

2419 

712 

2480 

254i 

2602 

2663 

2724 

2785 

2846 

2907 

2968 

3029 

7i3 

3090 

3i5o 

3211 

3272 

3333 

3394 

3455 

35i6 

3577 

3637 

7i4 

3698 

3759 

3820 

388i 

394i 

4OO2 

4o63 

4124 

4x85 

4245 

7i5 

43o6 

4367 

4428 

4488 

4549 

46io 

467o 

4?3i 

4792 

4852 

716 

4gi3 

4974 

5o34 

5095 

5i56 

52i6 

5277 

5337 

5398 

5459 

717 

55i9 

558o 

564o 

57oi 

576i 

5822 

5882 

5943 

6oo3 

6o64 

718 

6124 

6i85 

6245 

63o6 

6366 

6427 

6487 

6548 

6608 

6668 

60 

719 

6729 

6789 

685o 

6910 

697o 

7o3i 

-7091 

7i52 

7212 

7272 

N. 

0   |  1 

2 

3 

4 

5 

6 

7 

8 

9 

P. 

64  &  f  6. 

13 

J9 

26 

32 

38 

45 

5i 

58 

^ 

63  S    6 

i3 

J9 

25 

32 

38 

44 

5o 

5? 

£ 

62  M   6 

12 

I9 

25 

3i 

37 

43 

5o 

56 

5 

61  o  |   6 

12 

1  8 

24 

3i 

37 

43 

49 

55 

60  pL|  I  6 

12 

18 

24 

3o 

36 

4a 

48 

54 

10 


LOGARITHMS    OF    NUMBERS. 


i  N. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9  ID. 

720 

857332 

7393 

7453 

75i3 

7574 

7634  7694 

7755 

78i5 

7875 

60 

721 

7935 

7995 

8o56 

8116 

8i76 

8236 

8297 

8357 

84i7 

8477 

722 

8537 

8597 

8657 

87i8 

8778 

8838 

8898 

8958 

9018 

9c78 

723 

9i38 

9i98 

9258 

93i8 

9379 

9439 

9499 

9559 

96i9 

9679 

724 

9739 

9799 

9859 

9918 

9978 

..38 

..98 

.i58 

.218 

.278 

726 

86o338 

o398 

o458 

o5i8 

o578 

o637 

o697 

o757 

0817 

o877 

726 

o937 

0996 

io56 

1116 

ii76 

1236 

1295 

i355 

i4i5 

i475 

727 

i534 

i594 

i654 

1714 

I773 

i833 

i893 

I952 

2OI2 

2O72 

728 

2l3l 

2I9I 

225l 

23lO 

c37o 

243o 

2489 

2549 

26o8 

2668 

729 

2728 

2787 

2847 

29o6 

2966 

3o25 

3o85 

3i44 

3204 

3263 

73o 

3323 

3382 

3442 

35oi 

356i 

3620 

368o 

3739 

3799 

3858 

59 

73l 

39i7 

3977 

4o36 

4o96 

4i55 

4214 

4274 

4333 

4392 

4452 

782 

45x1 

457o 

463o 

4689 

4?48 

48o8 

4867 

4926 

4985 

5o45 

733 

5io4 

5i63 

5222 

5282 

534i 

54oo 

5459 

55i9 

5578 

5637 

734 

5696 

5755 

58i4 

5874 

SgSS 

6992 

6o5i 

6110 

6i69 

6228 

735 

6287 

6346 

64o5 

6465 

6524 

6583 

6642 

67oi 

676o 

68i9 

736 

6878 

6937 

6996 

7o55 

7114 

7i73 

7232 

•7291 

735o 

74o9 

737 

7467 

7526 

7585 

7644 

77o3 

•7-762 

•7821 

•7880 

7939 

7998 

738 

8o56 

8n5 

8i74 

8233 

8292 

835o 

84o9 

8468 

8527 

8586 

739 

8644 

87o3 

8762 

8821 

8879 

8938 

8997 

9o56 

9n4 

9i73 

74o 

9232 

9290 

9349 

94o8 

9466 

9525 

9584 

9642 

97oi 

976o 

74i 

98i8 

9877 

9935 

9994 

..53 

.in 

.1-70 

.228 

.287 

.345 

742 

87o4o4 

0462 

0621 

o579 

o638 

o696 

o755 

o8i3 

0872 

o93o 

58 

743 

o989 

io47 

1106 

ix64 

1223 

1281 

i339 

i398 

i456 

i5i5 

744 

i573 

i63i 

i69o 

i748 

l8o6 

i865 

I923 

1981 

2O4O 

2098 

745 

2i56 

22l5 

2273 

233i 

2389 

2448 

25o6 

2564 

2622 

2681 

746 

2739 

2797 

2855 

29i3 

2972 

3o3o 

3o88 

3i46 

3204 

3262 

747 

332i 

3379 

3437 

3495 

3553 

36ii 

3669 

3727 

3785 

3844 

748 

3902 

3960 

4oi8 

4o76 

4i34 

4l92 

425o 

43o8 

4366 

4424 

749 

4482 

454o 

4598 

4656 

47i4 

4772 

483o 

4888 

4945 

5oo3 

75o 

5o6i 

5u9 

5i77 

5235 

5293 

535i 

54o9 

5466 

5524 

5582 

75i 

564o 

5698 

5756 

58:3 

587i 

5929 

5987 

6o45 

6102 

6160 

762 

6218 

6276 

6333 

639i 

6449 

65o7 

6564 

6622 

6680 

6737 

753 

6795 

6853 

69io 

6968 

7O26 

7o83 

7i4i 

•7199 

7256 

73i4 

754 

737i 

7429 

7487 

7544 

•7602 

7659 

77i7 

7774 

7832 

•7889 

755 

7947 

8oo4 

8062 

8119 

8i77 

8234 

8292 

8349 

84o7 

8464 

5? 

766 

8522 

8579 

8637 

8694 

8752 

88o9 

8866 

8924 

898i 

9°39 

757 

9o96 

9i53 

92II 

9268 

9325 

9383 

944o 

9497 

9555 

9612 

758 

9669 

9726 

Q784 

984i 

q898 

9956 

..i3 

..70 

.I27 

.i85 

769 

880242 

O299 

o356 

o4i3 

o47i 

0028 

o585 

0642 

o699 

o756 

760 

o8i4- 

o87i 

0928 

o985 

IO42 

1099 

n56 

I2l3 

I27I 

i328 

761 

i385 

i442 

1499 

i556 

i6i3 

i67o 

I727 

i784 

i84i 

1898 

762 

i955 

2OI2 

2069 

2126 

2i83 

2240 

2297 

2354 

2411 

2468 

763 

2525 

258i 

2638 

2695 

2752 

2809 

2866 

2923 

2980 

3o37 

764 

3o93 

3i5o 

3207 

3264 

3321 

3377 

3434 

3491 

3548 

36o5 

765 

366i 

37i8 

3775 

3832 

3888 

3945 

4OO2 

4059 

4n5 

4l72 

•766 

4229 

4285 

4342 

4399 

4455 

45i2 

4569 

4625 

4682 

4739 

767 

4795 

4852 

4909 

4965 

5022 

5o78 

5i35 

5l92 

5248 

53o5 

768 

536i 

54i8 

5474 

553i 

5587 

5644 

57oo 

5757 

58i3 

587o 

769 

5926 

5983 

6o39 

6096 

6i52 

62O9 

6265 

632i 

6378 

6434 

56 

770 

6491 

6547 

66o4 

6660 

67i6 

6773 

6829 

6885 

6942 

6998 

771 

7o54 

7in 

7i67 

7223 

•7280 

7336 

7392 

7449 

75o5 

756i 

772 

76i7 

7674 

773o 

7786 

7842 

7898 

7955 

Son 

8o67 

8i23 

773 

8i79 

8236 

8292 

8348 

84o4 

846o 

85i6 

8573 

8629 

8685 

774 

874i 

8797 

8853 

8909 

8965 

9O2I 

9°77 

9i34 

9i9o 

9246 

N. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

D. 

60  £ 

6 

12 

18 

24 

3o 

36 

42 

48 

54 

£ 

59  $ 

6 

12 

18 

24 

3o 

35 

4i 

47 

53 

£• 

58  &. 

6 

12 

*7 

23 

29 

35 

4i 

46 

52 

S 

57  o 

6 

II 

'7 

23 

29 

34 

4o 

46 

5i 

56  £ 

6 

II 

'7 

22 

28 

34 

39 

45 

5o 

LOGARITHMS    OF    NUMBERS. 


N.    0 

1 

2 

3 

4     5 

6 

7 

8 

9 

D. 

775 

889802  !  9358 

94  1  4" 

9470 

9526 

9582 

9638 

9694 

975o 

9806 

5fi 

776 

9862 

9918 

9974 

..3o 

..86 

.141 

.197 

.253 

.3o9 

.365 

777 

890421 

0477 

o533 

0589 

o645 

o7oo 

0756 

0812 

0868 

0924 

778 

0080 

io35 

1091 

"4? 

1203 

1259 

i3i4 

i37o 

1426 

1482 

779 

i537 

i593 

1649 

1705 

i76o 

1816 

1872 

I928 

I983 

2039 

780 

2095 

2i5o 

2206 

2262 

23l7 

2373 

2429 

2484 

254o 

2595 

781 

265i 

2707 

2762 

2818 

2873 

2929 

2985 

3o4o 

3o96 

3i5i 

782 

3207 

3262 

33  1  8 

3373 

3429 

3484 

354o 

3595 

365i 

37o6 

783 

3762 

3817 

3873 

3928 

3984 

4039 

4o94 

4i5o 

42o5 

4261 

55 

7«4 

43i6 

4371 

4427 

4482 

4538 

4593 

4648 

47o4 

4759 

48i4 

7S5 

4870 

4925 

4980 

5o36 

5091 

5t46 

52OI 

5257 

53i2 

5367 

786 

5423 

5478 

5533 

5588 

5644 

5699 

5754 

58o9 

5864 

5920 

787 

5975 

6o3o 

6o85 

6i4o 

6195 

625i 

63o6 

636i 

64i6 

647i 

788 

6526 

658i 

6636 

6692 

6747 

6802 

6857 

69I2 

6967 

7O22 

789 

7°77 

7  1  32 

7187 

7242 

7297 

7352 

74o7 

•7462 

75i7 

7572 

79° 

•7627 

7682 

7737 

7792 

7847 

7902 

7957 

8012 

8o67 

8122 

79  l 

8176 

823i 

8286 

834i 

8396 

845  1 

85o6 

856i 

86i5 

867o 

792 

8725 

8780 

8835 

8890 

8944 

8999 

9o54 

9IO9 

9i64 

9218 

793 

0273 

9828 

9383 

9437 

9492 

9547 

96o2 

9656 

9711 

9766 

794 

9821 

9875 

993o 

9985 

..39 

..94 

.i49 

.203 

.258 

.312 

795 

900367 

0422 

0476 

o53i 

o586 

o64o 

o695 

o749 

0804 

oSSg 

796 

o9i3 

0968 

1022 

1077 

n3i 

1186 

1240 

I295 

i349 

i4o4 

797 

i458 

i5i3 

i567 

1622 

1676 

I73i 

i785 

1840 

i894 

i948 

54 

798 

2OO3 

2057 

2112 

2166 

2221 

2275 

2329 

2384 

2438 

2492 

799 

2547 

2601 

2655 

2710 

2764 

28l8 

2873 

2927 

298l 

3o36 

800 

3o9o 

3i44 

3i99 

3253 

33o7 

336i 

34i6 

3470 

3524 

3578 

801 

3633 

3687 

374i 

3795 

3849 

39o4 

3958 

4012 

4o66 

4120 

802 

4i74 

4229 

4283 

4337 

439i 

4445 

4499 

4553 

4607 

466  1 

8o3 

4716 

477° 

4824 

4878 

4932 

4986 

5o4o 

5o94 

5i48 

5202 

8o4 

5s56 

53io 

5364 

54i8 

5472 

5526 

558o 

5634 

5688 

5742 

8o5 

5798 

5fc5o 

59o4 

5958 

6012 

6066 

6n9 

6i73 

6227 

6281 

806 

6335 

6:189 

6443 

6497 

655i 

66o4 

6658 

6712 

6766 

6820 

8o7 

63  74 

6927 

6981 

7o35 

7089 

7i43 

7i96 

725o 

73o4 

7358 

808 

74n 

7465 

75l9 

7573 

7626 

768o 

7734 

7787 

784i 

7895 

809 

7949 

8002 

8o56 

8110 

8i63 

82I7 

8270 

8324 

8378 

843i 

810 

8485 

8539 

8592 

8646 

8699 

8753 

88o7 

8860 

89i4 

8967 

811 

902I 

9074 

9I28 

9i8i 

9235 

9289 

9342 

9396 

9449 

95o3 

812 

9556 

9610 

9663 

97i6 

9770 

9823 

9877 

993o 

9984 

..37 

53 

8i3 

9ioo9i 

oi44 

oi97 

025l 

o3o4 

o358 

o4n 

o464 

o5i8 

o57i 

8i4 

0624 

0678 

0731 

o784 

o838 

0891 

o944 

o998 

io5i 

no4 

8i5 

n58 

I2II 

1264 

i3i7 

i37i 

1424 

1477 

i53o 

1  584 

i637 

816 

i69o 

1743 

1797 

i85o 

1903 

i956 

2009 

2o63 

2116 

2i69 

817 

2222 

2275 

2328 

238i 

2435 

2488 

254i 

2594 

2647 

27OO 

818 

2753 

2806 

2859 

29l3 

2966 

3019 

3o72 

3i25 

3178 

323i 

819 

3284 

3337 

339o 

3443 

3496 

3549 

36o2 

3655 

37o8 

376i 

820 

38i4 

3867 

3920 

3973 

4026 

40-79 

4l32 

4i84 

4237 

4290 

821 

4343 

4396 

4449 

45o2 

4555 

46o8 

466o 

47i3 

4766 

48i9 

822 

4872 

4925 

4977 

5o3o 

5o83 

5i36 

5i89 

524i 

5294 

5347 

823 

5400 

5453 

55o5 

5558 

56n 

5664 

57i6 

5769 

5822 

5875 

824 

5927 

598o 

6o33 

6o85 

6i38 

6191 

6243 

6296 

6349 

64oi 

825 

6454 

65o7 

6559 

6612 

6664 

67i7 

677o 

6822 

6875 

6927 

826 

6980 

7o33 

7o85 

7i38 

7190 

7243 

7295 

7348 

74oo 

7453 

827 

75o6 

7558 

7611 

7663 

7716 

•7768 

•7820 

7873 

79s5 

7978 

5a 

828 

8o3o 

8o83 

8i35 

8188 

8240 

8293 

8345 

8397 

845o 

85o2 

829 

8555 

8607 

8659 

87I2 

8764 

8816 

8869 

892I 

8973 

9O26 

83o 

9078 

9i3o 

9i83 

9235 

9287 

9340 

9392 

9444 

9496 

9549 

N.  |   0 

1  |  2 

3 

4   [|  5 

6 

7 

8 

9 

P. 

.  f  55  3 

6 

ii 

J7 

22      28 

33 

39 

44 

5o 

1   54  1- 

5 

ii 

16 

22      27 

32 

38 

43 

49 

|    53  PH 

5 

ii 

16 

21      27 

3a 

37 

4a 

48 

«  I  52  p;  i  5 

10 

16 

21   II   26 

3i 

36 

4a 

4? 

18 


LOGARITHMS    OF    NUMDERS. 


N. 

0 

1 

2 

3 

4  ||   5 

6 

7 

8 

jQDr 

83i 

919601  9653 

9706 

9758 

9810 

9862 

99i4 

9967 

..i9 

..7i 

52   | 

832 

920123 

0176 

0228 

0280 

o332 

o384 

o436 

o489 

o54i 

o593 

833 

0645 

0697 

0749 

0801 

o853 

0906 

o958 

IOIO 

1062 

iix4 

834 

1166 

1218 

1270 

1322 

i374 

1426 

i478 

i53o 

i582 

i634 

835 

1686 

1738 

1790 

1842 

1894 

1946 

i998 

2o5o 

2IO2 

2i54 

836 

2206 

2258 

23lO 

2362 

24i4 

2466 

25i8 

2570 

2622 

2674 

837    27^6 

2777 

2829 

2881 

2933 

2985 

3o37 

3o89 

3i4o 

3i92 

838    3244 

3296 

3348 

3399 

345i 

35o3 

3555 

36o7 

3658 

37io 

839 

3762 

38i4 

3865 

39i7 

3969 

4O2I 

4072 

4i24 

4176 

4228 

84c 

4279 

433i 

4383 

4434 

4486 

4538 

4589 

464i 

4693 

4744 

84i 

4796 

4848 

4899 

495i 

5oo3 

5o54 

5io6 

5i57 

52O9 

5261 

842 

53i2 

5364 

54i5 

5467 

55i8 

557o 

562i 

5673 

5725 

5776 

843 

5828 

5879 

593i 

5982 

6o34 

•6o85 

6i37 

6188 

6240 

629I 

5i 

844 

6342 

6394 

6445 

6497 

6548 

6600 

665i 

6702 

6754 

68o5 

845 

6857 

6908 

6959 

7on 

•7062 

7u4 

7i65 

7216 

7268 

73i9 

346 

737o 

7422 

7473 

?524 

7576 

7627 

7678 

773o 

778i 

7832 

847 

7883 

7935 

7986 

8o37 

8088 

8i4o 

8i9i 

8242 

8293 

8345 

848 

8396 

8447 

8498 

8549 

8601 

8652 

87o3 

8754 

88o5 

8857 

849 

8908 

8959 

9oio 

9061 

9112 

9i63 

92l5 

9266 

93i7 

9368 

85o 

9419 

9470 

952I 

9572 

9623 

9674 

9725 

9776 

9827 

9879 

85i 

993o 

9981 

..32 

..83 

.i34 

.i85 

.236 

.287 

.338 

.389 

852 

93o44o 

0491 

o54a 

0592 

o643 

0694 

o745 

0796 

o847 

o898 

853 

0949 

IOOO 

io5i 

1  1  02 

u53 

1204 

1254 

i3o5 

i356 

i4o7 

854 

i458 

iSog 

i56o 

1610 

1661 

I7I2 

I763 

1814 

i865 

I9i5 

'  855 

1966 

2017 

2068 

2118 

2169 

222O 

227I 

2322 

2372 

2423 

856 

24?4 

2524 

2575 

2626 

2677 

2727 

2778 

2&29 

2879 

293o 

857 

2981 

3o3i 

3o82 

3i33 

3j83 

3234 

3285 

3335 

3386 

3437 

858 

3487 

3538 

3589 

3639 

369o 

374o 

379i 

384i 

3892 

3943 

859 

3993 

4o44 

4094 

4i45 

4i95 

4246 

4296 

4347 

4397 

4448 

*6o 

449B 

4549 

4599 

465o 

47oo 

475i 

48oi 

4852 

4902 

4953 

5o 

86  1 

5oo3 

5o54 

5io4 

5i54 

52o5 

5255 

53o6 

5356 

5406 

5457 

862 

55o7 

5558 

6608 

5658 

57o9 

5759 

58o9 

586o 

59io 

596o 

863 

6011 

6061 

6111 

6162 

6212 

6262 

63i3 

6363 

64i3 

6463 

864 

65i4 

6564 

66i4 

6665 

6715 

6765 

68i5 

6865 

69i6 

6966 

865 

7016 

7066 

7117 

7167 

7217 

?267 

73i7 

7367 

74i8 

7468 

866 

75i8 

7568 

7618 

7668 

77i8 

7769 

7819 

7869 

79i9 

7969 

867 

8019 

8069 

8119 

8169 

8219 

8269 

8320 

837o 

8420 

847o 

868 

85ao 

857o 

8620 

8670 

8720 

8770 

8820 

887o 

8920 

897o 

869 

9020 

9070 

9120 

9i7o 

9220 

9270 

9320 

9369 

94i9 

9469 

870 

95l9 

9569 

9619 

9669 

97I9 

9769 

98i9 

9869 

99i8 

9968 

871 

940018 

0068 

0118 

0168 

0218 

0267 

o3i7 

o367 

0417 

o467 

872 

o5i6 

o566 

0616 

0666 

o7i6 

o765 

o8i5 

o865 

o9i5 

o964 

873 

1014 

io64 

in4 

n63 

I2l3 

1263 

i3i3 

i362 

1412 

1462 

874 

i5n 

i56i 

1611 

1660 

1710 

i76o 

i8o9 

i859 

I9o9 

i958 

875 

2008 

2o58 

2IO7 

2l57 

22O7 

2256 

23o6 

2355 

24o5 

2455 

876 

25o4 

2554 

26o3 

2653 

2702 

2752 

2801 

285i 

29OI 

295o 

877 

3ooo 

3o49 

3099 

3i48 

SigS 

3247 

3297 

3346 

3396 

3445 

49 

878 

3495 

3544 

3593 

3643 

3692 

3742 

379i 

384i 

389o 

3939 

879 

3989 

4o38 

4o88 

4i37 

4i86 

4236 

4285 

4335 

4384 

4433 

!  880 

4483 

4532 

458i 

463i 

468o 

4729 

4779 

4828 

4877 

4927 

881 

4976 

5o25 

5o74 

6124 

5i73 

5222 

5272 

532i 

537o 

54i9 

882 

5469 

55i8 

5567 

56i6 

5665 

57i5 

5764 

58:3 

5862 

5912 

883 

596i 

6010 

6059 

6108 

6i57 

6207 

6256 

63o5 

6354 

64o3 

884 

6452 

65oi 

655i 

6600 

6649 

6698 

6747 

6796 

6845 

6894 

885 

6943 

6992 

7o4i 

7o9o 

7i4o 

7189 

7238 

7287 

7336 

7385 

886 

7434 

7483 

7532 

758i 

763o 

7679 

7728 

7777 

•7826 

7875 

N. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

D. 

{62  «  ,'  5 

10 

16 

21 

26 

3i 

36 

4a 

4? 

5i  si  5 

10 

i5 

20 

26 

3i 

36 

4i 

46 

50  *' 

5 

IO 

i5 

2O 

25 

3o 

35 

4o 

45 

49  * 

K 
ij 

10 

i5 

20 

25 

29 

34 

39 

44 

LOGARITHMS    OF    NUMBERS. 


10 


K 

0 

1 

2 

3 

4  ||   5 

6 

7 

8 

9 

I). 

887 

947924 

7973 

8022 

8070 

8119 

8168  8217 

8266 

83i5 

8364 

49 

888 

84i3 

8462 

85u 

856o 

8609 

8657 

8706 

8755 

88o4 

8853 

889 

8902 

895i 

8999 

9048 

9097 

9i46 

9i95 

9244 

9292 

934i 

890 

939° 

9439 

9488 

9536 

9585 

9634 

9683 

973i 

9780 

0820 

891 

9878 

9926 

9975 

..24 

..73 

.  121 

.170 

.2I9 

.267 

.3i6 

892 

95o365 

o4i4 

0462 

o5n 

o56o 

0608 

o657 

0706 

0754 

o8o3 

893 

o85i 

o9oo 

0949 

0997 

io46 

io95 

n43 

II92 

I24o 

I289 

894 

i338 

i386 

i435 

i483 

i532 

i58o 

i629 

1677 

1726 

1775 

895 

1823 

1872 

1920 

1969 

2017 

2066 

2Il4 

2i63 

2211 

2260 

43 

896 

23o8 

a356 

24o5 

2453 

2502 

255o 

2599 

2647 

2696 

2744 

897 

2792 

284  1 

2889 

2938 

2986 

3o34 

3o83 

3i3i 

3i8o 

3228 

898 

3276 

3325 

3373 

3421 

3470 

35j8 

3566 

36i5 

3663 

3711 

899 

3760 

38o8 

3856 

39o5 

3953 

4ooi 

4o49 

4o98 

4i46 

4i94 

900 

4243 

4291 

4339 

4387 

4435 

4484 

4532 

458o 

4628 

4677 

901 

4726 

4773 

4821 

4869 

49i8 

4966 

5oi4 

5o62 

5uo 

5i58 

902 

6207 

5355 

53o3 

535i 

5399 

5447 

5495 

5543 

5592 

564o 

903 

5688 

5736 

5784 

5832 

588o 

5928 

5976 

6024 

6072 

6120 

904 

6168 

6216 

6265 

63i3 

636i 

64o9 

6457 

65o5 

6553 

6601 

906 

6649 

6697 

6745 

6793 

684o 

6888 

6936 

6984 

7032 

7080 

906 

7128 

7176 

7224 

7272 

7320 

7368 

74i6 

7464 

75i2 

7559 

907 

7607 

7655 

77o3 

775i 

7799 

7847 

7894 

7942 

799° 

8o38 

908 

8086 

8i34 

8181 

8229 

8277 

8325 

8373 

8421 

8468 

85i6 

909 

8564 

8612 

8659 

8707 

8755 

88o3 

885o 

8898 

8946 

8994 

910 

9041 

9089 

9i37 

9i85 

9232 

928o 

9328 

9375 

9423 

9471 

911 

95i8 

9566 

9614 

9661 

97°9 

9757 

98o4 

9852 

9900 

9947 

912 

9995 

..42 

..90 

.i38 

.1.85 

.233 

.280 

.328 

.376 

.423 

9i3 

96o47i 

o5i8 

o566 

o6i3 

0661 

o7o9 

0756 

08  o4 

o85i 

0899 

914 

o946 

o994 

io4i 

1089 

u36 

1184 

I23l 

I279 

i326 

i374 

47 

916 

1421 

1469 

i5i6 

i563 

1611 

i658 

1706 

i753 

1801 

1  848 

916 

i895 

i943 

i99o 

2038 

2o85 

2132 

2180 

2227 

2275 

2322 

917 

2369 

2417 

2464 

25ll 

2559 

2606 

2653 

27OI 

2748 

2795 

918 

2843 

289o 

2937 

2985 

3o32 

3o79 

3i26 

3i74 

3221 

3268 

919 

33i6 

3363 

34io 

3457 

35o4 

3552 

3599 

3646 

3693 

374i 

920 

3788 

3835 

3882 

3929 

3977 

4024 

4071 

4n8 

4i65 

4212 

921 

4260 

43o7 

4354 

44oi 

4448 

4495 

4542 

4590 

4637 

4684 

922 

473i 

4778 

4825 

4872 

49i9 

4966 

5oi3 

5o6i 

5io8 

5i55 

923 

5202 

5249 

5296 

5343 

539o 

5437 

5484 

553i 

5578 

5625 

924 

5672 

57i9 

5766 

58i3 

586o 

59o7 

5954 

6001 

6o48 

6095 

926 

6142 

6i89 

6236 

6283 

6329 

6376 

6423 

6470 

65i7 

6564 

926 

6611 

6658 

67o5 

6752 

6799 

6845 

6892 

6939 

6986 

7o33 

927 

7080 

7127 

7i73 

7220 

7267 

73i4 

736i 

74o8 

7454 

75oi 

928 

7548 

7595 

7642 

7688 

7735 

7782 

7829 

7875 

7922 

7969 

929 

8016 

8062 

8io9 

8i56 

82o3 

8249 

8296 

8343 

839o 

8436 

93o 

8483 

853o 

8576 

8623 

8670 

8716 

8763 

8810 

8856 

8903 

93i 

895o 

8996 

9o43' 

9090 

9i36 

9i83 

9229 

9276 

9323 

9369 

982 

94i6 

9463 

95o9 

9556 

96o2 

9649 

9695 

9742 

9789 

9835 

933 

9882 

9928 

9975 

.  .21 

..68 

.n4 

.161 

.207 

.254 

.3oo 

934 

97o347 

o393 

o44o 

o486 

o533 

o579 

0626 

0672 

°7r9 

0765 

46 

935 

0812 

o858 

0904 

o95i 

°997 

io44 

io9o 

ii37 

u83 

1229 

936 

1276 

1322 

i369 

i4i5 

i46i 

i5o8 

i554 

1601 

1  647 

i693 

937 

1740 

1786 

i832 

1879 

I925 

i97i 

2018 

2064 

2110 

2157 

938 

2203 

2249 

2295 

2342 

2388 

2434 

248  1 

2527 

2573 

2619 

939 

2666 

2712 

2758 

2804 

285i 

2897 

2943 

2989 

3o35 

3o82 

940 

3i28 

3|74 

3220 

3266 

33i3 

3359 

34o5 

345  1 

3497 

3543 

94r 

359o 

3636 

3682 

3728 

3774 

3820 

3866 

SgiS 

3959 

4oo5 

942 

465  j 

4097 

4i43 

4i89 

4235 

4281 

4327 

4374 

4420 

4466 

943 

45i2 

4558  46o4 

465o 

4696 

4742 

4788 

4834 

488o 

4926 

N.  |   0 

1 

2 

3 

4   ||  5 

6 

7 

8 

9 

D. 

g  f  43  »  f  5 

10 

i4 

J9 

a4 

29 

34 

38 

43 

1   47  -  '!   5 

9 

i4 

'9 

24 

28 

33 

38 

42 

P  1  46  p;    5 

9 

r4 

1  9 

23 

28 

32 

37 

4i 

20 


LOGARITHMS    OF    NUMBERS. 


N. 

0 

1 

2 

3    4     5 

6 

7 

8 

9   II 

944 

9?4Q72 

5oi8 

5o64 

5no 

5i56 

5202 

5248 

5294 

534o 

5386 

46 

945 

5432 

5478 

5524 

557o 

56i6 

5662 

5707 

5753 

5799 

5845 

946 

589i 

5937 

5983 

6o29 

6075 

6121 

6167 

6212 

6258 

63o4 

9^7 

635o 

6396 

6442 

6488 

6533 

6579 

6625 

6671  6717 

6763 

948 

6808 

6854 

69oo 

6946 

6992 

7°37 

7083 

7i29  7175 

7220 

949 

7266 

7312 

7358 

74o3 

7449 

7495 

754i 

7586 

7632 

7678 

95c 

7724 

7769 

78i5 

7861 

79o6 

795a 

7998 

8o43 

8o89 

8i35 

961 

8181 

8226 

8272 

83i7 

8363 

84o9 

8454 

85oo 

8546 

859i 

952 

8637 

8683 

8728 

87?4 

88i9 

8865 

8911 

8956 

9OO2 

9o47 

953 

9°93 

9i38 

9i84 

923o 

9275 

932I 

9366 

94i2 

9457 

95o3 

9^4 

9548 

9594 

9639 

9685 

973o 

9776 

9821 

9867 

99I2 

9958 

955 

98ooo3 

oo49 

oo94 

oi4o 

oi85 

023l 

0276 

0322 

0367 

O4l2 

45 

966 

o458 

o5o3 

o549 

o594 

o64o 

o685 

0730 

0776 

0821 

0867 

957 

O9I2 

o957 

ioo3 

1048 

io93 

1139 

1184 

I229 

1275 

1320 

958 

i366 

1411 

i456 

i5oi 

i547 

l592 

i637 

i683 

1728 

1773 

959 

1819 

i864 

I9°9 

i954 

20OO 

2o45 

2090 

2i35 

2181 

2226 

960 

2271 

23i6 

2362 

2407 

2452 

2497 

2543 

2588 

2633 

2678 

961 

2723 

2769 

2814 

2859 

29o4 

2949 

2994 

3o4o 

3o85 

3i3o 

962 

3175 

322O 

3265 

33io 

3356 

34oi 

3446 

349i 

3536 

358i 

963 

3626 

367i 

37i6 

3762 

38o7 

3852 

3897 

3942 

3987 

4o32 

964 

4077 

4l22 

4i67 

4212 

4257 

4302 

4347 

4392 

4437 

4482 

965 

4527 

4572 

46i7 

4662 

4707 

4?52 

4797 

4842 

4887 

4932 

966 

4977 

5O22 

5067 

5lI2 

5i57 

52O2 

5247 

5292 

5337 

5382 

967 

5426 

5471 

55i6 

556i 

56o6 

565i 

5696 

574l 

5786 

583o 

968 

5875 

592O 

5965 

6010 

6o55 

6100 

6i44 

6i89 

6234 

6279 

969 

6324 

6369 

64i3 

6458 

65o3 

6548 

6593 

6637 

6682 

6727 

97° 

6772 

6817 

6861 

69o6 

695i 

6996 

7040 

7o85 

7i3o 

7i75 

971 

7219 

7264 

73°9 

7353 

7398 

7443 

7488 

7532 

7577 

7622 

972 

7666 

7711 

7756 

7800 

7845 

7890 

7934 

7979 

8024 

8068 

973 

8n3 

8i57 

8202 

8247 

829I 

8336 

838i 

8425 

8470 

85i4 

974 

8559 

86o4 

8648 

8693 

8737 

8782 

8826 

8871 

8916 

896o 

975 

9oo5 

9o49 

9o94 

9i38 

9i83 

9227 

9272 

93i6 

936i 

94o5 

976 

945o 

9494 

9539 

9583 

9628 

9672 

97i7 

976i 

98o6 

985o 

44 

977 

9895 

9939 

9983 

..28 

..72 

.117 

.161 

.206 

.25o 

•294 

978 

99o339 

o383 

0428 

0472 

o5i6 

o56i 

o6o5 

o65o 

o694 

0738 

979 

o783 

0827 

0871 

o9i6 

0960 

ioo4 

io49 

io93 

n37 

1182 

980 

1226 

1270 

i3i5 

i359 

i4o3 

i448 

l492 

i536 

i58o 

1625 

981 

i669 

1713 

1758 

1802 

1  846 

1890 

i935 

i979 

2O23 

2067 

982 

21  I  I 

2i56 

22OO 

2244 

2288 

2333 

2377 

2421 

2465 

25o9 

983 

2554 

2598 

2642 

2686 

2730 

2774 

28l9 

2863 

2907 

295l 

984 

2995 

3o39 

3o83 

3127 

3172 

32i6 

3260 

33o4 

3348 

3392 

985 

3436 

348o 

3524 

3568 

36i3 

3657 

3701 

3745 

3789 

3833 

986 

3877 

392I 

3965 

4oo9 

4o53 

4o97 

4i4i 

4i85 

4229 

4273 

987 

43i7 

436i 

44o5 

4449 

4493 

4537 

458i 

4625 

4669 

47'3 

988 

4757 

48oi 

4845 

4889 

4933 

4977 

5021 

5o65 

5io8 

5i52 

989 

5i96 

524o 

5284 

5328 

5372 

54i6 

5460 

55o4 

5547 

559i 

99° 

5635 

5679 

5723 

5767 

•58i  i 

5854 

5898 

5942 

5986 

6o3o 

99i 

6074 

6117 

6161 

62o5 

6249 

6293 

6337 

638o 

6424 

6468 

992 

65i2 

6555 

6599 

6643 

6687 

6731 

6774 

6818 

6862 

69o6 

993 

6949 

6993 

7o37 

7080 

7124 

7168 

7212 

7255 

7299 

7343 

994 

7386 

743o 

7474 

7517 

756i 

7605 

7648 

7692 

7736 

7779 

995 

7823 

7867 

79io 

7954 

7998 

8o4i 

8o85 

8l29 

8172 

8216 

996 

8259 

83o3 

8347 

839o 

8434 

8477 

852i 

8564 

8608 

8652 

997 

8695 

8739 

8782 

8826 

8869 

89i3 

8956 

9ooo 

9043 

9087 

99s 

9i3i 

9i74 

92l8 

926l 

93o5 

9348 

9392 

9435 

9479 

9522 

999 

9565 

96o9 

965a 

9696 

9739 

9783 

9826 

987o 

99i3 

9957 

43 

N. 

0 

1    2 

3 

4  !|  5 

6 

7 

8 

9 

D._ 

.  f  46  a  f   5 

9 

i4 

18  I!  23 

28 

32 

37 

4i 

fel  45  is  I   5 

9 

i4 

18     23 

27 

32 

36 

4i 

|   44  *   4 

9 

i3 

l8      22 

26 

3i 

35 

4o 

fi  I  43  fc  I  4 

9 

i3 

I7       22 

26 

3o 

34 

39 

1 

TABLE 


OF 


LOGARITHMIC  SINES  AID  TANGENTS 


FOR    EVERY 


TEN  SECONDS  OF  THE  QUADRANT. 


22 


LOGARITHMIC    SINES. 


Win. 

Sine  of  0  Degree. 

P.  Part 

0" 

10" 

20" 

30'' 

40" 

50" 

to  1". 

O 

Inf.  Neg. 

5.685575 

5.9866o5 

6.  162696 

6.287635 

6.384545 

59 

I 

6.463726 

6.53o673 

6.588665 

6398i7 

685575 

•726968 

58 

2 

764756 

799518 

83i7o3 

861666 

889695 

916024 

57 

3 

940847 

964328 

986605 

7.007794 

7.027997 

7.o473o3 

56 

4 

7.065786 

7-o835i5 

7.ioo548 

116939 

i32733 

i47973 

55 

5 

162696 

176936 

190725 

204089 

2i7o54 

229643 

54 

6 

241877 

253776 

265358 

276639 

287635 

298358 

53 

7 

3o8824 

319043 

329027 

338787 

348332 

357672 

52 

8 

3668i6 

37577i 

384544 

393145 

401678 

409850 

5i 

9 

417968 

425937 

433762 

44i449 

449002 

456426 

5o 

10 

463726 

470904 

477966 

4849i5 

49i754 

498488 

49 

689.4 

1  1 

5o5n8 

5  i  1649 

5i8o83 

524423 

53o672 

536832 

48 

629.4 

12 

542906 

548897 

5548o6 

56o635 

566387 

572o65 

4? 

e 

579.1 

i3 

577668 

5832oi 

588664 

594059 

599388 

6o4652 

46 

536.2 

i4 

6o9853 

614993 

620072 

625093 

63oo56 

634964 

45 

499.2 

i5 

6398i6 

6446  i  5 

64936i 

654o56 

6587oi 

66329-7 

44 

467.0 

16 

667845 

672345 

676799 

681208 

685573 

689895 

43 

438.7 

17 

694i73 

698410 

702606 

706762 

7io879 

714907 

42 

4x3.6 

18 

718997 

722999 

726965 

730896 

734791 

73865i 

4i 

391.3 

'9 

742478 

746270 

75oo3i 

753758 

757455 

761  119 

4o 

371.2 

20 

764754 

768358 

771932 

775477 

778994 

782482 

39 

353.1 

21 

785943 

789376 

792782 

796162 

7995i5 

802843 

38 

336.7 

22 

806146 

809423 

812677 

816906 

819111 

822292 

3? 

321.7 

23 

82545i 

828586 

831700 

834791 

83786o 

84090-7 

36 

3o8.o 

24 

843934 

846939 

849924 

852889 

855833 

858757 

35 

295.4 

25 

861662 

864548 

8674i5 

870262 

873092 

8-75902 

34 

283.8 

26 

878695 

881470 

884228 

886968 

889690 

892396 

33 

273.1 

27 

895o85 

897758 

900414 

9o3o54 

905678 

90828-7 

32 

263.2 

28 

910879 

9i3457 

916019 

9i8566 

921098 

923616 

3i 

254.0 

29 

926119 

928608 

931082 

933543 

935989 

938422 

3o 

245.4  1 

3o 

940842 

943248 

94564i 

948020 

950387 

95274i 

29 

237.5 

3i 

955082 

957411 

959727 

962031 

964322 

966602 

38 

229.8 

32 

968870 

971126 

97337o 

9756o3 

977824 

980034 

27 

222.7 

33 

982233 

984421 

986598 

988764 

990919 

993064 

26 

216.1 

34 

995198 

997322 

999435 

8.ooi538 

3.oo363i 

8.oo57i4 

25 

209.8 

35 

8.007787 

8.009850 

8.011903 

013947 

016981 

oi8oo5 

24 

2o3.9 

36 

020021 

022027 

024023 

026011 

02-7989 

029959 

23 

198.3 

37 

oSigig 

033871 

o358i4 

037749 

039676 

041^92 

22 

IQ3.0 

38 

o435oi 

o454oi 

047294 

049178 

o5io54 

052922 

2  I 

188.0 

39 

054781 

o56633 

o58477 

o6o3i4 

062142 

o63963 

20 

i83.s 

4o 

065776 

067582 

069380 

071171 

o72955 

o7473i 

J9 

178.7 

4i 

076500 

078261 

080016 

081764 

o835o4 

o85238 

1  8 

i74.4 

42 

o86965 

088684 

090398 

092104 

093804 

096497 

i? 

170.3 

43 

097183 

098863 

100537 

IO22O4 

io3864 

io55i9 

16 

166.4 

44 

107167 

108809 

no444 

II2O74 

n3697 

Il53i5 

i5 

162.6 

45 

116926 

n8532 

I2Ol3l 

121725 

I233i3 

124895 

i4 

iSg.  i 

46 

126471 

128042 

129607 

i3ii66 

l3272O 

134268 

i3 

i55.6 

4? 

i358io 

i37348 

138879 

i4o4o6 

i4i927 

143443 

12 

i52.4 

49 

144953 

146458 

i47959 

149453 

150943 

152428 

II 

149.2 

49 

iWqoy 

i55382 

156852 

i583i6 

i59776 

i6i23i 

10 

146.2 

5o 

162681 

164126 

165566 

167002 

i68433 

169859 

9 

i43.3 

5! 

171280 

172697 

174109 

I755i7 

1-76920 

1-78319 

8 

i4o.5 

52 

179713 

i8no3 

182488 

183869 

i85245 

186617 

7 

i37-9 

53 

i87985 

i89348 

190707 

192062 

I934i3 

194760 

6 

i35.3 

54 

196102 

197440 

198774 

20OIO4 

2oi43o 

202752 

5 

i32.8 

55 

204070 

2o5384 

206694 

2O8OOO 

209302 

210601 

4 

i3o.4 

56 

2ii895 

2i3i85 

214472 

2i5755 

2i7o34 

218309 

3 

128.1 

57 

219681 

220849 

2221  l3 

223374 

224631 

225884 

2 

125.9 

58 

227134 

22838o 

229622 

23o86i 

232096 

233328 

I 

123.7 

59 

234557 

235782 

237003 

238221 

239.436 

24o647 

O 

121.  6 

80" 

50"         40" 

30"         20" 

10" 

Co-sine  of  S9  Degrees. 

Min. 

LOGARITHMIC    TANGENT  s. 


Min. 

Tangent  of  0  Degree. 

0' 

10" 

20" 

30-'' 

40''    |    50" 

o 

Inf.  Neg. 

5.685575 

5.986605 

6.  162696 

6.287635 

6.384545 

« 

6.463726 

6.53o673 

6.588665 

6398l7 

686676 

726968 

2 

764756 

7995i8 

83i7o3 

861666 

889696 

9i6o24 

3 

940847 

964329 

9866o5 

7.007794 

7  027998 

7.047808 

A 

7.065786 

7.o835i5 

7.ioo548 

116939 

i32733 

147973 

5 

162696 

i76937 

190726 

204089 

217064 

229643 

6 

241878 

253777 

265359 

276640 

287635 

298869 

7 

3o8825 

319044 

329028 

338788 

348333 

357673 

8 

366817 

3V5772 

384546 

393146 

401679 

409862 

9 

4i797° 

426989 

433764 

44i45i 

449004 

466428 

10 

463727 

470906 

477968 

484917 

491766 

498490 

ii 

5o5i2o 

5n65i 

5i8o85 

624426 

530676 

536835 

.*2 

542909 

548900 

5548o8 

56o638 

666890 

672068 

13 

577672 

583204 

588667 

694062 

699891 

6o4655 

14 

609857 

614996 

620076 

626097 

680060 

634968 

i5 

639820 

644619 

649366 

654o6i 

668706 

663302 

16 

667849 

672350 

6768041   68i2i3 

686678 

689900 

ll 

694179 

698416 

702612 

706768 

710886 

714963 

18 

719003 

723oo5 

726972 

730902 

734797 

788668 

'9 

742484 

746277 

760087 

753765 

767462 

761127 

20 

764761 

768365 

771940 

775485 

779002 

782490 

21 

785951 

789384 

79279° 

796170 

799624 

802862 

22 

8o6i55 

8o9433 

812686 

816916 

819120 

822802 

23 

825460 

828596 

831710 

8348oi 

887870 

840918 

24 

843944 

846950 

849935 

862900 

855844 

868769 

25 

861674 

86456o 

867426 

870274 

878104 

876916 

26 

878708 

88i483 

884240 

886981 

889704 

892410 

27 

895o99 

897772 

900428 

9o3o68 

906692 

908801 

28 

910894 

913471 

916034 

918681 

921118 

92363i 

29 

926134 

928623 

93io98 

933559 

936oo6 

938439 

3o 

94o858 

943265 

945658 

948037 

95o4o4 

952758 

Si 

955ioo 

957428 

969745 

962049 

96434i 

96662i 

32 

968889 

971145 

973389 

976622 

977844 

98oo54 

33 

982253 

98444i 

9866i8 

988786 

99o94o 

993o85 

34 

995219 

997343 

999467 

8.  001660 

8.003653 

8.006736 

35 

8  007809 

8.009872 

8,oi  1926 

013970 

016004 

oi8o29 

36 

020044 

O22o5l 

024048 

O26o35 

028014 

029984 

37 

o3i945 

033897 

o3584o 

037776 

089701 

041618 

38 

043527 

045428 

047321 

049206 

061081 

o52949 

39 

054809 

o56662 

068606 

060342 

062171 

o63992 

4o 

o658o6 

067612 

069410 

071201 

072986 

074761 

4? 

o7653i 

078293 

080047 

081796 

o83536 

086270 

4a    086997 

088717 

090431 

092187 

098887 

o9553o 

43 

097217 

098897 

100671 

102289 

108900 

io5554 

44 

107203 

io8845 

110481 

II2IIO 

118784 

ii5352 

45 

116963 

118569 

120169 

I2I763 

i2335i 

i24933 

46 

i265io 

128081 

129646 

i3i2o6 

182760 

i  343o8 

47 

i3585i 

i37389 

138921 

i4o447 

141969 

i43485 

48 

144996 

i465oi 

i48ooi 

149497 

160987 

162472 

49 

i53952 

155426 

166896 

i5836i 

169821 

161276 

5o 

162727 

164172 

i656i3 

167049 

168480 

169906 

5i 

171328 

172745 

i74i58 

176666 

176969 

i78368 

52 

i79763 

i8u53 

182538 

183919 

186296 

186668 

53 

i88o36 

189400 

I9o76o 

192116 

198466 

194818 

54 

196156 

197494 

198829 

200169 

2oi485 

202808 

55 

204126 

206440 

2o675o 

208067 

209869 

210668 

56 

211953 

213243 

2i453o 

216814 

217098 

218869 

57 

219641 

220909 

222I74 

223434 

224692 

>25945 

58 

227195 

228442 

229686 

280924 

282160 

233392 

59 

234621 

235846 

237o68 

238286 

289602 

240718 

60"        50"        40" 

30'' 

20" 

10" 

Co-tangent  of  89  Degrees. 

P.  Tart 

tol". 

~5g 

58 

57 

56 

55 

54 

53 

62 

5i 

5o 

49 

689.4 

48 

629.4 

47 

679.1 

46 

536.2 

45 

499.2 

44 

467.0 

43 

438.7 

42 

4i3.6 

4  1 

39i.3 

4o 

371.2 

q 

353.1 

38 

336.7 

37 

321.7 

36 

3o8.o 

35 

295.4 

34 

283.  9 

33 

278.2 

32 

268.2 

3i 

264.0 

3o 

245.4 

29 

287.8 

28 

229.8 

27 

222.7 

26 

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32I6 

3546 

21 

39 

4968 

54?,3 

5877633i 

6784 

7237 

2O 

39 

3876 

4206 

4536 

4866 

5i95 

5524 

20 

4o 

7689 

8i4i 

85929043 

9494 

9944 

19 

4o 

5852 

6181 

65og 

6837 

7i65 

7492 

'9 

4i 

8.870393 

0842 

1291  i739 

2187 

2634 

18 

4i 

7819 

8t  46 

8473 

8799 

9126 

945i 

18 

42 

3o8o 

3527 

3972  44i8 

4863 

53o7 

17 

42 

9777 

.io3 

.428 

.753 

10-78 

1402 

17 

43 

575i 

6194 

6638!7o8o 

7522 

7964 

16 

43 

8.811726 

2o5o 

2374 

2698 

3021 

3344 

16 

44 

84o5 

8846 

9286^9726 

.166 

.6o5 

i5 

44 

3667 

3989 

43i2 

4634 

4956 

5277 

i5 

45 

8  68io43 

i48i 

191912356 

2793 

323o 

i4 

45 

5599 

592O 

6241 

656i 

6882 

7202 

i'4 

46 

3665 

4ioi 

45364971 

54o5 

5838 

i3 

46 

7522 

7841 

8161 

848o 

8799 

9Il8 

i3 

4? 

6272 

67o5 

71377669 

8001 

8432 

12 

47 

9436 

;755 

..73 

.390 

.-708 

1025 

12 

48 

8863 

9293 

9723  |.i52 

.58i 

IOIO 

II 

48 

8.82i343 

i659 

1976 

2292 

2609 

2925 

II 

49 

8.  69i438 

1866 

2293 

272O 

3i46 

3572 

IO 

49 

324o 

3556 

387i 

4i86 

45OI 

48:6 

IO 

5o 

3998 

4423 

4848 

5272 

5696 

6120 

9 

5o 

5i3o 

5444 

5758 

6072 

6385 

6698 

9 

5i 

6543 

6966 

7388 

78io 

8232 

8653 

8 

5i 

•7011 

7324 

7637 

7949 

8261 

8573 

8 

52 

9o73 

9494 

9913 

.333 

.752 

1171 

7 

52 

8884 

9196 

95°7 

9818 

.  129 

•  439 

7 

53 

8.701589 

2OO7 

2424 

2841 

3258 

3674 

6 

53 

8.83o749 

1060 

1369 

1679 

1988 

2298 

6 

54 

4090 

45o5 

4920 

5335 

5  749 

6i63 

5 

54 

26o7 

29l5 

3224 

3532 

384o 

4i48 

5 

55 

6577 

6990 

7402 

78i5 

8226 

8638 

4 

55 

4456 

4763 

5o7o 

5377 

5684 

599i 

4 

56 

9049 

9460 

987o 

.280 

.690 

1099 

3 

56 

629-7 

66o3 

6909 

72i5 

752O 

7825 

3 

5? 

8,7n5o7 

1916 

2324 

273i 

3:39 

3546 

2 

57 

8i3o 

8435 

874o 

9044 

9348:9652 

2 

58 

3g52 

4358 

4764 

5i69 

5574 

5979 

I 

58 

9956 

.260 

.563 

.866 

1169 

l472 

I 

J.9 

6383 

6787 

7190 

7593 

7996 

3398 

0 

59 

8.84r774 

2O76 

2378 

2680 

2982 

3283 

O 

60" 

OOP 

40" 

30" 

20" 

10" 

c 

60" 

50"  |  40" 

30"  |  20" 

10" 

c 

Co-sine  of  87  Degrees. 

a 

Co-sine  of  86  Degrees. 

S 

C  j//2,.  3"  4"  5"  6"  7"  8"  9" 
•J  48  96  145  1S3  241  289  338  386  434 

.(  1"  2"  3"  4"  5"  G"  7"  8"  9" 
irt}  34  69  103  138  172  207  2-11  275  310 

LOGARITHMIC    TANGENTS. 


.5 

Tangent  of  2  Degrees. 

_c 

Tangent  of  3  Degrees. 

s 

0" 

10" 

20" 

30"   40" 

50" 

§ 

0" 

10" 

20" 

30" 

40"  j  50" 

o 

6.543o84 

3687 

4289 

4891 

5492 

6092 

59 

0 

8.yi9396 

9798 

.2OI 

.6o3 

ioo4 

i4o5 

59 

i 

6691 

7289 

7887 

8483 

9079 

9674 

58 

i 

8.72i8o6 

22O7 

2607 

3007 

3406 

38o5 

58 

2 

8  550268 

0862 

i454 

2046 

2637 

3227 

57 

2 

42o4 

4602 

5ooo 

5397 

5794 

6191 

^7 

3 

38i7 

44o5 

4993 

558o 

6166 

6752 

56 

3 

6588 

6984 

738o 

7775 

8170 

8565 

56 

4 

y336 

7920 

85o3 

9085 

9667 

.248 

55 

4 

8959 

9353 

9746 

.  i4o 

.533 

.925 

55 

5 

8.56o828 

1407 

1985 

2563 

3i4o 

37i6 

54 

5 

8.73i3i7 

I709 

2101 

2492 

2883 

3273 

54 

6 

4291 

4866 

544o 

6oi3 

6585 

7l57 

53 

6 

3663 

4o53 

4442 

483i 

5220 

56o8 

53 

7 

7727 

8298 

8867 

9435 

...3 

.570 

52 

7 

5996 

6384 

6771 

7i58 

7545 

793i 

52 

6 

8.57n37 

1702 

2267 

2832 

3395 

3958 

5i 

8 

83i7 

87o3 

9088 

9473 

9858 

.242 

5i 

9 

4520 

5o8i 

5642 

6201 

6760 

73i9 

5o 

9 

8.740626 

1009 

i393 

1776 

2i58 

254o 

5o 

10 

7877 

8434 

8990 

9545 

.  IOO 

.654 

49 

10 

2922 

33o4 

3685 

4o66 

4447 

4827 

49 

ii 

8-  58  1  208 

1760 

23l2 

2864 

34i4 

3964 

48 

ii 

5207 

5586 

5966 

6344 

6723 

7101 

48 

12 

45i4 

5o62 

56io 

6i57 

6704 

7249 

47 

12 

7479 

7857 

8234 

8611 

8988 

9364 

47 

i3 

7795 

8339 

8883 

9426 

9968 

.5io 

46 

i3 

974o 

.116 

•  49i 

.866 

1241 

i6i5 

40 

i4 

8.5gio5i 

i5gi 

2l3l 

2670 

3208 

3746 

45 

i4 

8.751989 

2363 

2736 

3109 

3482 

3855 

45 

i5 

4283 

4820 

5355 

5890 

6425 

6959 

44 

i5 

4227 

4599 

497o 

534i 

5712 

6o83 

44 

16 

7492 

8024 

8556 

9087 

9618 

.i4? 

43 

16 

6453 

6823 

7i93 

7562 

793i 

83oo 

43 

17 

8.600677 

I2O5 

1733 

2260 

2787 

33i3 

42 

ll 

8668 

9o36 

94o4 

9771 

.139 

.5o5 

42 

18 

3839 

4363 

4887 

54n 

5934 

6456 

4i 

18 

8.760872 

1238 

1604 

1970 

2335 

2700 

4i 

^9 

6978 

7499 

8019 

8539 

go58 

9576 

4o 

J9 

3o65 

3429 

3793 

4i57 

4520 

4884 

4o 

20 

8.610094 

0612 

1128 

1  644 

2160 

2675 

39 

20 

5246 

5609 

597i 

6333 

6695 

7o56 

39 

21 

3189 

3702 

42i5 

4728 

524o 

575i 

38 

21 

74i7 

7778 

8139 

8499 

8859 

9218 

38 

22 

6262 

6772 

7281 

779° 

8298 

8806 

37 

22 

9578 

9937 

.295 

.654 

IOI2 

1370 

37 

23 

93i3 

9819 

.325 

.83o 

i335 

i839 

36 

23 

8.77i727 

2o85 

2442 

2798 

3i55 

35n 

36 

24 

8.622343 

2846 

3348 

385o 

435i 

4852 

35 

24 

3866 

4222 

4577 

4932 

5287 

564i 

35 

25 

5352 

585i 

635o 

6849 

7346 

7844 

34 

25 

5995 

6349 

6702 

7o56 

7409 

7761 

34 

26 

834o 

8836 

9332 

9827 

.321 

.8i5 

33 

26 

8n4 

8466 

8817 

9169 

9520 

9871 

33 

27 

8.63i3o8 

1801 

2293 

2785 

3276 

3766 

32 

27 

8.780222 

o572 

0922 

1272 

1622 

1971 

32 

28 

4256 

4746 

5235 

5723 

6211 

6698 

3i 

28 

2320 

2669 

3017 

3365 

37i3 

4o6i 

3i 

29 

7i84 

76?I 

8i56 

864i 

9126 

9610 

3o 

29 

44o8 

4755 

5  I  02 

5448 

5794 

6i4o 

3o 

3o 

8.640093 

0576 

io58 

i54o 

2O2I 

25O2 

29 

3o 

6486 

683i 

7177 

7521 

7866 

8210 

29 

3i 

2982 

3462 

394i 

4420 

4898 

5376 

28 

3i 

8554 

8898 

9242 

9585 

9928 

.271 

28 

32 

5853 

6329 

68o5 

7281 

7756 

823o 

27 

32 

8.79o6i3 

0955 

1297 

1639 

1980 

2321 

27 

33 

8704 

9178 

965i 

.123 

.595 

1067 

20 

33 

2662 

3oo3 

3343 

3683 

4023 

4362 

26 

34 

8.65i537 

2008 

2478 

2947 

34x6 

3884 

25 

34 

47oi 

5o4o 

5379 

57i8 

6o56 

6394 

25 

35 

4352 

4820 

5286 

5753 

6219 

6684 

24 

35 

673i 

7o69 

74o6 

7743 

8079 

84i6 

24 

36 

7149 

7613 

8077 

854i 

9004 

9466 

23 

36    8752 

9088 

9423 

9759 

..94 

.429 

23 

37 

9928 

.389 

.85o 

i3n 

1771 

2230 

22 

37 

8.8oo763 

1098 

i432 

i765 

2099 

2432 

22 

38 

8.662689 

3i48 

36o6 

4o63 

4520 

4977 

21 

38 

2765 

3098 

343i 

3763 

4095 

4427 

21 

39 

5433 

5889 

6344 

6799 

7253 

7707 

20 

39 

4758 

Sogo 

5421 

575i 

6082 

6412 

2O 

4o 

8160 

86i3 

9065 

95i7 

9968 

.419 

'9 

4o 

6742 

7072 

7402 

773i 

8060 

8389 

'9 

4i 

8.670870 

1320 

1769 

2218 

2667 

3n5 

18 

4i 

87i7 

9O46 

9374 

9701 

..29 

.356 

18 

4s 

3563 

4oio 

4457 

4903 

5349 

5794 

17 

42 

8.8io683 

IOIO 

i337 

i663 

1989 

23  1  5 

i? 

43 

6239 

6684 

7128 

7572 

8oi5 

8457 

16 

43 

2641 

2966 

329I 

36i6 

394i 

4265 

16 

44 

8900 

934i 

9783 

.224 

.664 

uo4 

i5 

44 

4589 

4913 

5237 

556o 

5884 

6207 

i5 

45 

8.68i544 

1983 

2422 

2860 

3298 

3735 

i4 

45 

6529 

6852 

7174 

7496 

7818 

8Uo 

i4 

46 

4172 

46o8 

5o44 

548o 

59i5 

635o 

i3 

46 

846  1 

8782 

9io3 

9423 

9744 

..64 

i3 

47 

6784 

7218 

7652 

8o85 

85i7 

895o 

12 

47 

8.820384 

o7o3 

1023 

1  342 

1661 

1980 

I  2 

48 

9381 

9813 

.244 

•  674 

no4 

i534 

II 

48 

2298 

26l7 

2935 

3253 

3570 

3888 

II 

49 

8.691963 

2392 

2820 

3248 

3675 

4io3 

10 

49 

42o5 

4522 

4838 

5i55 

547i 

5787 

IO 

5o 

4529 

4956 

538i 

5807 

6232 

6656 

9 

5o 

6io3 

64i8 

6733 

7049 

7363 

7678 

9 

5i 

7081 

75o4 

7928 

835i 

8773 

9i95 

8 

5i 

7992 

83o7 

8621 

8934 

9248 

9661 

8 

52 

9617 

..38 

.459 

.880 

i3oo 

1720 

7 

52 

9874 

.187 

.5oo 

.812 

1124 

i436 

7 

53 

8.702139 

2558 

2976 

3395 

38i2 

4230 

6 

53 

8.83i748 

2o5g 

2371 

2682 

2992 

33o3 

6 

54 

4646 

5o63 

5479 

5895 

63io 

6725 

5 

54 

36i3 

3924 

4234 

4543 

4853 

5i62 

5 

55 

7140 

7554 

7967 

838i 

8794 

9206 

4 

55 

547i 

578o 

6o89 

6397 

67o5 

7013 

4 

56 

9618 

..3o 

.442 

.853 

1263 

i674 

3 

56 

732I 

7629 

7936 

8243 

855o 

8857 

3 

57 

8.712083 

2493 

2902 

33n 

37i9 

4127 

2 

57 

9i63 

9470 

9776 

..81 

.387 

.692 

2 

58 

4534J4942 

5348 

5755 

6161 

6567 

I 

58 

8.840998 

i3o3 

1607 

I9I2 

2216 

2521 

I 

59 

6972 

7377 

7781 

8186 

8589 

8993 

O 

59 

2825 

3i28 

3432 

3735 

4o38 

434l 

O 

60" 

50"   40" 

30" 

20" 

10" 

fi 

60" 

50" 

40" 

30" 

20" 

10" 

B* 

Co-tangent  of  87  Degrees. 

•9 

S 

Co-tangent  of  86  Degrees. 

P  PnrtJ  1//2//  3//  4"   5"   fi"   7//   8"   9" 

"*j  48  97  145  193  242  290  338  387  435 

(  1»2"  3"  i"  5"  6"  7"  8"  9;/ 
m\  35  69  104  138  173  207  242  275  311 

LOGARITHMIC    SINES. 


J 

Sine  of  4  Degrees. 

y 

Sine  of  5  Degrees. 

* 

0" 

10" 

20" 

30" 

40"  ]  50" 

i 

0" 

10" 

20" 

30"  '  40" 

50" 

o 

8.843585 

3886 

4i86 

4487 

4787 

5o87 

59 

o 

8.94o296 

o537 

°777 

IOI7 

1258 

i498 

59 

I 

5387 

5687 

5987 

6286 

6585 

6884 

58 

I 

I73« 

i977 

22I7 

2696 

2935 

58 

2 

7i83 

748  1 

7780 

8078 

8376 

8673 

57 

2 

3i74 

34i3 

3652 

SSgi 

4i29 

4368 

57 

3 

897i 

9268 

9565 

9862 

.i59 

.455 

56 

3 

46o6 

4844 

5o83 

532i 

5558 

5796 

56 

4 

8.85o75i 

1047 

1  343 

1639 

i934 

2229 

55 

4 

6o34 

627I 

65o8 

6745 

6982 

72I9 

55 

5 

2525 

28l9 

3ii4 

34o8 

37o3 

3997 

54 

5 

7456 

7693 

7929 

8166 

8402 

8638 

54 

6 

429I 

4584 

4878 

5i7i 

5464 

55io 

53 

6 

8874 

9no 

9345 

958i 

9817 

..52 

53 

7      OO^Q 

6342 

6634 

6926 

72l8 

52 

7 

8.95o287 

0522 

o757 

o992 

1227 

i46i 

53 

8    78oi 

8o92 

8383 

8674 

8965 

9255 

5i 

8 

i696 

I93o 

2164 

2398 

2632 

2866 

5i 

9 

9546 

9836 

.126 

.4i5 

.7o5 

994 

5o 

9 

3ioo 

3333 

3567 

38oo 

4o33 

4266 

5o 

10 

8.86i283 

1572 

1861 

2149 

2438 

2-726 

49 

10 

4499 

4732 

4965 

5i97 

5429 

5662 

49 

ii 

3oi4 

33o2 

3589 

3877 

4i64 

445  1 

48 

ii 

5894 

6126 

6358 

659o 

6821 

7o53 

48 

12 

4738 

5o24 

53n 

5597 

5883 

6i69 

47 

12 

7284 

75i6 

7747 

7978 

82O9 

844o 

47 

i3 

6455 

6740 

7025 

73io 

7595 

788o 

46 

i3 

8670 

8901 

9i3i 

9362 

9592 

9822 

46 

i4 

8i65 

8449 

8733 

901-7 

93oi 

9585 

45 

i4 

8.96oo52 

0282 

o5n 

0741 

o97o 

I2OO 

45 

15 

9868 

.i5i 

.434 

.7i7 

1000 

1282 

44 

i5 

1429 

i658 

1887 

2116 

2344 

2573 

44 

16 

8.87i565 

1847 

2129 

2692  29^3 

43 

16 

2801 

3o3o 

3258 

3486 

37i4 

3942 

43 

i7 

3255 

3536 

38i7'4o97 

4378 

4658 

42 

'7 

4170 

4397 

4625 

4852 

5o8o 

53o7 

42 

18 

4938 

52i8 

5498 

5777 

6o57 

6336 

4i 

18 

5534 

576i 

5987 

6214 

644  1 

6667 

4i 

i9 

66i5 

6894 

7172 

745  1 

7729 

8oo7 

4o 

i9 

6893 

7I2O 

7346 

7572 

7797 

8023 

4o 

20 

8285 

8563 

884i 

9118 

9395 

9672 

39 

20 

8249 

8474 

8700 

8925 

9i5o 

9375 

39 

21 

9949 

.226 

.5o3 

•779 

io55 

i33i 

38 

21 

96oo 

9825 

..49 

.274 

•  498 

.723 

38 

22 

8.881607 

i883 

2i58 

2433 

2-708 

2983 

37 

22 

8.97o947 

n7i 

i395 

i6i9 

1842 

2066 

37 

23 

3258 

3533 

38o7 

4o8i 

4355 

4629 

36 

23 

2289 

25i3 

2736 

2959 

3i82 

34o5 

36 

24 

49o3 

5i77 

5996 

6269 

35 

24 

3628 

385i 

4o73 

4296 

45i8 

4740 

35 

25 

6542 

68i4 

7087  7359 

763i 

79°3 

34 

25 

4962 

5i84 

54o6 

5628 

585o 

6071 

34 

26 

8174 

8446 

8717 

8988 

9259 

953o 

33 

26 

6293 

65i4 

6735 

6956 

7177 

7398 

33 

27 

98oi 

..71 

.34i 

.612 

.882 

n5i 

32 

27 

7619 

7839 

8060 

8280 

85oi 

8721 

3a 

28  8.891421 

i69o 

1960 

2229 

2498 

2767 

3i 

28 

894i 

9i6i 

938i 

96oo 

9820 

..39 

3i 

29    3o35 

33o4 

3572!384o 

4io8 

4376 

3o 

29 

8.98o259 

o478 

o697 

0916 

n35 

i354 

3o 

3o 

4643 

49n 

5i78;5445 

57I2 

5979 

29 

3o 

i573 

i79i 

2OIO 

2228 

2447 

2665 

29 

3i 

6246 

65i2 

6778  7044 

73io 

7576 

28 

3i 

2883 

3ioi 

33  1  9 

3536 

3754 

3972 

28 

82 

7842 

8107 

8373,8638 

89o3 

9i68 

27 

32 

4i89 

44o6 

4623 

484o 

5o57 

5274 

27 

33 

9432 

9697 

9961 

.225 

•  489 

.753 

26 

33 

549i 

57o8 

5924 

6i4i 

6357 

6573 

26 

348.90101-7 

1280 

1  544  1807 

2O70 

2333 

25 

34 

6789 

7oo5 

7221 

7437 

7652 

7868 

25 

35 

2596 

2858 

3i2i  3383 

3645 

3907 

24 

35 

8o83 

8299 

85i4 

8729 

8944 

9l59 

24 

36 

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4692  4953 

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36 

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37 

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6257  65i7 

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37 

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2610 

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49 

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10 

49 

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7661 

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52 

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7 

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1052 

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6 

5!3 

9.oio737 

0942 

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6 

54 

1  5  '±L 

i789 

2o35 

2280 

2525 

2770 

5 

5^ 

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2i65 

2369 

2572 

2776 

2979 

5 

55 

3oi5 

3260 

35o4 

3749 

3993 

4 

55 

3i82 

3385 

3588 

379i 

3994 

4i97 

4 

56 

448  1 

4725 

4969 

5212 

5456 

5699 

3 

56 

44oo 

4602 

48o5 

5007 

3 

*7 

5942 

6i85 

6428 

667i 

69i4 

7i56 

2 

57 

56i3 

58i5 

6017 

6219 

6421 

6622 

2 

58 

7398 

7641 

7883 

8i25 

8366 

8608 

I 

58 

6824 

7025 

7227 

7428 

-7629 

783o 

I 

59 

885o 

9o9I 

9332 

9573 

98i4 

..55 

O 

59 

8o3i 

8232 

8433 

8633 

8834 

9o34 

O 

60" 

50" 

40" 

30" 

20" 

10" 

c 

60" 

50" 

40" 

30" 

20" 

10" 

E° 

Co-sine  of  85  Degrees. 

9 

Co-sine  of  84  Degrees. 

i 

Tt  p   (1"  2"  3"  4"  5"  6"  7"  8"  9" 

p  p  .<  1"  2"  3"  4"  5"  6"  7"  8"  9"  j 

1   irt\  27  53  80  107  134  160  187  214  241 

irt\  22  44  GO  87  109  131  153  175  1!)7 

LOGARITHMIC    TANGENTS. 


29 


1 

Tangent  of  4  Degrees. 

d 

Tangent  of  5  Degrees. 

m 

(X 

10" 

20" 

30" 

40" 

50" 

's 

0" 

10" 

20" 

30" 

40" 

50" 

0 

8.844644 

4946 

5248 

555i 

5852 

6i54 

59 

o 

8.94i952 

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2437 

2679 

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6455 

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58 

i 

34o4 

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3888 

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2 

8260 

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2 

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1260 

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2738 

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4 

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1784 

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8 

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3677 

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62 

8.93oi55 

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7 

52 

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199-7 

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2411 

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53 

1  647 

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2639 

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6 

53 

3o3i 

3237 

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365o 

3856 

4062 

6 

54 

3i34 

338i 

3629 

3876 

4123 

4369 

5 

54 

4268 

4474 

468o 

4886 

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5297 

5 

55 

46i6 

4862 

5io9 

5355 

56oi 

5847 

4 

55 

55o2 

57o7 

59i3 

6118 

6323 

6528 

4 

56 

6o93 

6339 

6584 

683o 

7o75 

7320 

3 

56 

6732 

6937 

7l42 

7346 

755i 

7755 

3 

57 

7565 

7810 

8o55 

8299 

8544 

8788 

2 

57 

7959 

8i64 

8368 

8572 

8776 

8979 

2 

58 

9o32 

9276 

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9.764 

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I 

58 

9i83 

9387 

959o 

9794 

9997 

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I 

59 

8.94o49^ 

o738 

o98i 

1224 

i467 

1700. 

o 

59 

9.o2o4o3 

0606 

o8o9 

IOI2 

I2l5 

i4i8 

O 

60" 

50" 

40" 

30" 

20" 

10" 

60" 

50" 

40" 

30" 

'  20"   10" 

a 

Co-tangent  of  85  Degrees. 

.3 
S 

Co-tangent  of  84  Degrees. 

1 

p  Po  .<  1"  2"  3"  4"  5"  6"  7"  8"  9" 
irl\  27  54  81  108  135  162  188  215  242 

.<  1"  2"  3"  4"  5"  6"  7"  8"  9"  \ 
I  22  44  66  88  110  132  154  177  199 

LOGARITHMIC    SINES. 


1  — 

.9 

S 

tSine  of  6  Degrees. 

1 

Sine  of  7  Degrees. 

0" 

10" 

20" 

30" 

40"   W 

0" 

10" 

20"  |  30" 

40"  1  50" 

019.019235 

9435 

9635 

9835 

..35 

.235 

59 

O 

9.085894 

6066 

6237 

64o9 

658o 

675i 

59 

I  9.020435 

o635 

o834 

io34 

1233 

i433 

58 

I 

6922 

7093 

7264 

7435 

7606 

7777 

58 

a|    i632 

i83i 

2o3o 

2229 

2428 

2627 

57 

2 

7947 

8118 

8288 

8459 

8629 

8800 

57 

1  ^ 

2825 

3o24 

3223 

342i 

36i9 

38i8 

56 

3 

8970 

9140 

93io 

948o 

965i 

982O 

56 

4 

4oi6 

4214 

44i2 

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48o7 

5oo5 

55 

4 

999° 

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.839 

55 

5    52o3 

54oo 

5598 

5795 

5992 

6189 

54 

5 

9.091008 

1178 

1  347 

i5i6 

i685 

i855 

54 

fij    6386 

6583 

6780 

6977 

7174 

7370 

53 

6 

2024 

2193 

2362 

253o 

2609 

2868 

53 

7 

7367 

7763 

7960 

8i56 

8352 

8548 

52 

7 

3o37 

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3374 

3542 

o  * 
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8 
9 

8744 
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9i36 
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9332 
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9527 
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9723 

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5i 
5o 

8 
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4047 
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4216 

5223 

4384 
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4552 
5559 

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10 

ty.o3io89 

1284 

1479 

1673 

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6062 

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6898 

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2839 

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7566 

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48 

li: 

3421 

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12 

8066 

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16 

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16 

2048 

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42 

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4a 

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29    2/49 

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793i 

8090 

8249 

84o8 

27 

33 

7172 

7356 

7539  7722 

7905 

8088 

26 

33 

8567 

8726 

8884 

9o43 

9202 

9360 

26 

34 

827I 

8454 

86378820 

9002 

9i85 

25 

34 

95l9 

9677 

9836 

9994 

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25 

35    9367 

955o 

97329914 

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.278 

24 

35 

9.120469 

0627 

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1259 

24 

36  9.060460 

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1188 

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23 

36 

1417 

i574 

I732 

1890 

2047 

2205 

23 

37 

i55i 

I732 

1914  2095 

2276 

2457 

22 

37 

2362 

2520 

2677 

2835 

2992 

3i49 

22 

38 

2639 

2820 

3ooi|3i8i 

3362 

3543 

21 

38 

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3463 

3620 

3777 

3934 

4091 

21 

39 

3724 

3904 

4o85'4265 

4445 

4626 

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39 

4248 

44o4 

456i 

4718 

48  74 

5o3i 

2O 

4o 

48o6 

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5526 

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4o 

5187 

5344 

55oo 

5656 

58i2 

5969 

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4i 

5885 

6o65 

6244  6424 

66o3 

6783 

18 

4i 

6i25 

6281 

6437 

6593 

6748 

6904 

18 

42 

6962 

7i4i 

7320  7499 

7678 

7857 

'7 

42 

7060 

7216 

737i 

7527 

7682 

7838 

17 

43 

8o36 

82i5 

8393,85728751 

8929 

16 

43 

7993 

8i49 

83o4 

84^9 

86i4 

877o 

16 

44 

9I07 

9286 

9464  9642  9820 

9998 

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44 

8925 

9o8o 

9235 

939o 

9544 

9699 

i5 

459.070176 

o354 

o532  0709  0887 

io65 

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45 

9854 

...9 

.i63 

.3i8 

-472 

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i4 

46    1242 

1420 

1597  1774 

1951 

2128 

i3 

46 

9.i3o78i 

o936 

io9o 

1244 

i398 

i552 

i3 

47 

23o6 

2482 

2659 

2836 

3oi3 

3i9o 

12 

4? 

I7o6 

1860 

20l4 

2168 

2322 

2476 

12 

48 

3366 

3543 

37i9 

3896 

4072 

4248 

II 

48 

263o 

2783 

2937 

3091 

3244 

3398 

II 

49 

4424 

46oo 

4777 

4952 

5i28 

53o4 

10 

49 

355i 

3704 

3858 

4on 

4x64 

43i7 

IO 

5o 

548o 

5656 

583i 

6007 

6182 

6358 

9 

5o 

447o 

4623 

4776 

4929 

5o82 

5235 

9 

5i 

6533 

6708 

6883 

7o58 

7233 

74o8 

8 

5i 

5387 

554o 

5693 

5845I5998 

6i5o 

8 

52 

7583 

7758 

7933 

8107 

8282 

8457 

7 

52 

63o3 

6455 

66o7 

6760(6912 

7064 

7 

53 

863i 

88o5 

8980 

9i54 

9328 

95o2 

6 

53 

7216 

7368 

7520 

7672 

7824 

7976 

6 

54 

9676 

985o 

..24 

.198 

.372 

.545 

5 

54 

8128 

8279 

843i 

8582 

8734 

8886 

5 

55 

9.o8o7i9 

0892 

1066 

1239 

i4i3 

i586 

4 

55 

9o37 

9i88 

934o 

9491 

9642 

9793 

4 

56j    i759 

1932 

2IO5 

2278 

245i 

2624 

3 

56 

9944 

..96 

.247 

.398 

.548 

.699 

3 

57 

2797 

2369 

3:42 

33:4 

3487 

3659 

2 

57 

9.:4o85o 

IOOI 

n5i 

1,302 

i453 

i6o3 

2 

58 

3832 

4oo4 

4176 

4348 

45so 

4692 

I 

58 

1754 

1904(2055 

22O5 

2355 

25o5 

I 

59 

4864 

5o36 

5208 

538o 

555i 

5723 

0 

59 

2655 

280612956 

3io6 

3256]34o5 

O 

60" 

50" 

40" 

30" 

20" 

10" 

c 

60*     50"  |  40"   30"   20"   1C-" 

d 

Co-sine  of  83  Degrees. 

§ 

Co-sine  of  82  Degrees.     8 

.  C  1"  2"  3"  4"  5"  6"  7"  8"  9" 

p  p  A  I"  2"  3"  4"  5"  6"  7"  8"  9" 

T»  P&Trt  <         - 

irt}  16  32  48  64  80  96  112  128  144 

LOGARITHMIC    TANGENTS. 


31 


1 

Tangent  of  6  Degrees. 

.5 

Tangent  of  7  Degrees. 

0" 

10"  |  20" 

30" 

40" 

50'' 

0" 

10" 

20" 

30" 

40" 

50" 

0 

9.021620 

1823 

2025 

2227 

2430 

2632 

59 

o 

9.089144 

9318 

9492 

9666 

9839 

..i3 

59 

I 

2834 

3o36 

3238 

3439 

364i 

3843 

58 

i 

9.090187 

o36i 

o534 

0708 

0881 

io54 

58 

2 

4o44 

4245 

4447 

4648 

4849 

5o5o 

57 

2 

1228 

i4oi 

i574 

I747 

1920 

2093 

57 

3 

525i 

5452 

5653 

5853 

6o54 

6254 

56 

3 

2266 

2439 

2612 

2784 

2957 

3129 

56 

4    6455 

6655 

6855 

7o55 

7255 

7455 

55 

4 

33o2 

3474  3647 

38i9 

3991 

4i63 

55 

5 

7655 

7855 

8o55 

8254 

8454 

8653 

54 

5 

4336 

45o8 

468o 

485i 

5o23 

5i95 

54 

6 

8852 

go52 

925i 

945o 

9649 

9848 

53 

6 

5367 

5538 

5710 

588i 

6o53 

6224 

53 

7.9.030046 

0245 

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0841 

1039 

52 

7 

6395 

6567 

6738 

6909 

7o8o 

725l 

5s 

8f   i237 

i435 

i633 

i83i 

2029  2227 

5i 

8 

7422 

7593 

7764 

7934 

8io5 

8276 

5i 

91    2425 

2623 

2#20 

3oi7 

32i534i2 

5o 

9 

8446 

8616 

8787 

8957 

912-7 

9298 

5o 

10 

3609 

38o6 

4oo3 

4200 

4397 

4594 

49 

IO 

9468 

9638 

9808 

9978 

.i48 

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49 

ii 

4791 

4987 

5i84 

538o 

5576 

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48 

ii 

9.  100487 

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0827 

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1166 

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48 

12 

5969 

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636i 

6557 

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6948 

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12 

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2OI2 

2181 

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47 

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7i44 

7339 

7535 

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7926 

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46 

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2519 

2688 

2857 

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8706 

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9o95 

9290 

45 

i4 

3532 

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45 

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9485 

9679 

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44 

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4542 

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52i4 

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44 

16 

9.o4o65i 

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1039 

1232 

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16 

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2007 

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2394 

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42 

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18 

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18 

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8060 

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4i 

19 

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4322 

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856o 

8727 

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9060 

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20 

5284 

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5668 

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6243 

39 

20 

9559 

9726 

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39 

21 

6434 

6626 

6817 

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7200 

739i 

38 

21 

q.  no556 

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1219 

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38 

22 

7682 

7773 

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8346 

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37 

22 

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1882 

2047 

2213 

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37 

23 

8727 

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9108 

9298 

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36 

23 

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2873 

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3204 

3368 

36 

24 

9869 

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35 

24 

3533 

3698 

3863 

4028 

4l92 

4357 

35 

25 

9.o5ioo8 

1197 

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i576 

1766 

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34 

25 

452i 

4686 

485o 

5oi5 

5i79 

5343 

34 

26 

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2522 

27II 

2900 

3o88 

33 

26 

55o7 

567i 

5835 

5999 

6i63 

6327 

33 

27 

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3654 

3843 

4o3i 

4219 

32 

27 

6491 

6655 

6818 

6982 

7i45 

73°9 

32 

28 

4407 

4596 

4784 

4972 

5i5g 

5347 

3i 

28 

7472 

7636 

7799 

7962 

8126 

8289 

3i 

29 

5535 

5723 

59io 

6o98 

6285 

6472 

3o 

29 

8452 

86!5 

8778 

894i 

9104 

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29 

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29 

3i 

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834i 

8528  8714 

28 

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9.120404 

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28 

32 

8900 

9086 

9273 

9459 

9645  983i 

27 

32 

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i863 

2025 

2187 

27 

33 

9.060016 

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0759  0944 

26 

33 

2346 

25io 

2671 

2833 

2994 

3i56 

26 

34 

n3o 

i3i5 

i5oo 

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1870  2o55 

25 

34 

33i7 

3478 

364o 

38oi 

3962 

4123 

25 

35 

2240 

2425 

2610 

2795 

2979  3i64 

24 

35 

4284 

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24 

36 

3348 

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23 

36 

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23 

37 

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4637 

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22 

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6532 

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22 

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21 

38 

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21 

39 

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18 

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42 

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ll 

42 

0994 

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1628 

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43 

9.071027 

1208 

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1751 

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16 

43 

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2IO2 

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2419 

2577 

2735 

16 

44 

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2294 

2475 

2655 

2836  3oi6 

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44 

2893 

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3366 

3524 

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45 

3i97 

3377 

3558 

3738 

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45 

3839 

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4i54 

43l2 

4469 

4626 

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46 

4278 

4458 

4638 

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46 

4784 

4941 

5098 

5255 

54i2 

5569 

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4? 

5356 

5536 

57i5 

5895 

607416253 

12 

47 

5726 

5883 

6o4o 

6l97 

6353 

65io 

12 

48 

6432 

6611 

679o 

6969 

7i48 

7327 

II 

48 

6667 

6823 

6980 

7I36 

7292 

7449 

II 

49 

75o5 

7684 

7862 

8o4i 

8219 

8398 

IO 

49 

76o5 

776i 

-7918 

8o74 

823o 

8386 

IO 

5o 

8576 

8754 

8932 

9IIO 

9288 

9466 

9 

5o 

8542 

8698 

8854 

9009 

9i65 

9321 

9 

5i 

9644 

9822 

.  .  .  . 

.177 

.355 

.532 

8 

5i 

9476 

9632 

9788 

9943 

..98 

.254 

8 

62 

9.080710 

0887 

1064 

1241 

1419 

i596 

7 

52 

9.140409 

o564 

0720 

o875 

io3o 

i:85 

7 

53 

I773 

1950 

212,6 

23o3 

2480 

2657 

6 

53 

i34o 

i495 

i65o 

i8o5 

I959 

2Il4 

6 

54 

2833 

3oio 

3i86 

3362 

3539 

37i5 

5 

54 

2269 

2424 

2578 

2733 

2887 

3o42 

5 

55 

3891 

4o67 

4243 

44i9 

4595 

477i 

4 

55 

3196 

335o 

35o4 

3659 

38i3 

3967 

4 

56 

4947 

5l22 

5298 

5473 

5649 

5824 

3 

56 

4l2I 

4275 

4429 

4583 

4737 

4890 

3 

5? 

6000 

6i75 

635o 

6525 

6700 

6875 

2 

57 

5o44 

5i98 

535i 

55o5 

5659 

58i2 

2 

58 

7o5o 

7225 

74oo 

7574 

7749 

7924 

I 

58 

5966 

6119 

6272 

6425 

6579 

6732 

I 

59 

8098 

8273 

8447 

8621 

8795 

897o 

0 

59 

6885 

7o38 

7191 

7344 

7497 

765o 

0 

60" 

50" 

40" 

30" 

20" 

30" 

a 

60" 

50" 

40" 

30" 

20" 

10" 

• 

Co-tangent  of  83  Degrees. 

ffl 

& 

Co-tangent  of  82  Degrees. 

1 

P  Part  5  r/  2"  3"  4//  5"  6//  7"  8"  9"  !!«,   ,<  I"  2"  3"  4"  5"  6"  7"   8"  9" 
iri\  19  37  56  75  94  112  131  150  163  j    irt{  16  33  49  G5  81  98  114  130  14fi 

LOGARITHMIC    SINES. 


g  1      Sine  of  d  Degrees. 

.3 

Sine  of  9  Degrees. 

*  p  o77   f~io" 

W  [  30" 

40" 

50" 

^ 

0"    |  10" 

20" 

30" 

40" 

'  w 

~o;9i  143555 

37'>5 

3855 

4oo5 

4i54 

43o4 

59 

o 

9.  194332 

4465 

4598 

473i 

4864 

4997 

59 

I 

4453 

46o3 

4?52 

4902 

5o5i 

5200 

58 

I 

5129 

5262 

5395 

5527 

566o 

5792 

58 

2 

5349 

5498 

5648 

5797 

5946^095 

57 

2 

5925 

6o57 

6i89 

6322 

6454 

6586 

57 

3    6243|6392J654i 

6690 

68396987 

56 

3 

6719 

685i  6983 

7n5 

7247 

7379|56 

4    7i36 

7284!7433  758i 

773o 

7878 

55 

4 

75u 

7643 

7775 

7907 

8o38 

8i7o!55 

5!    8026 

8i74 

832318471 

8619 

8767 

54 

5 

8302 

8434 

8565 

8697 

8828 

8960 

54 

6    89i5 

9063 

921119358 

95o6 

9654 

53 

6 

9091 

9223 

9354 

9486 

96l7 

9748 

53 

7j    98o2 

99^9!  "97 

.244 

.392 

.539 

52 

7 

9879 

.  .11 

.142 

.273 

•  4o4 

.535 

52 

8 

9.  160686 

083410981 

1128 

1275 

1422 

5i 

8 

9.200666 

0797 

0928 

io59 

1189 

1320 

5i 

9 

i569 

I7i6 

i863 

2OIO 

2l57 

23o4 

5o 

9 

i45i 

i582 

1712 

i843 

i973 

2104 

5o 

10 

a45i 

2597 

2744 

2891 

3o37 

3i84 

49 

10 

2234 

2365 

2495 

2626 

2756 

2886 

49 

ii 

333o 

3476 

3623 

3769 

39i5 

4o6i 

48 

ii 

3017 

3i47 

3277 

34o7 

3537 

3667 

48 

12 

4208 

4354 

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4646 

4792 

4938 

47 

12 

3797 

3927 

4o57 

4187 

43i7 

4447 

47 

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5o83 

5229 

5375  552i 

5666 

58i2 

46 

i3 

4577 

4706 

4836 

4966 

5o95 

5225 

46 

i4 

5957 

6io3 

6248  6394 

6539 

6684 

45 

i4 

5354 

5484 

56i3 

5743 

5872 

6002 

45 

i5 

683o 

6975 

7120  7265 

74io 

7555 

44 

i5 

6i3i 

6260 

6389 

6019 

6648 

6777 

44 

16    7700 

7845 

799o8i35 

8279 

8424 

43 

16 

6906 

7o35 

7164 

7293 

7422 

755i 

43 

i7 

8569 

87i3 

8858  9002 

91^7 

9291 

42 

17 

7679 

7808 

7937 

8066 

8194 

8323 

42 

18 

9435 

gSSo 

97249868 

.  .12 

.i56 

4i 

18 

8452 

858o 

8709 

8837 

8966 

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4i 

19 

9.  i6o3oi 

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0589  0732 

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9479 

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20 

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39 

20 

9992 

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39 

21 

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2169 

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2599 

2742 

38 

21 

9.210760 

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38 

22 

2885 

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3i72  33i5 

3458 

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37 

22 

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1  654 

1781 

1909 

2037 

2164 

37 

23 

3743 

3886 

40294172 

43i4 

4457 

36 

23 

2291 

2419 

2546 

2674 

2801 

2928 

36 

24 

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4742 

4885,5o27 

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35 

24 

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3437 

3564 

3691 

35 

25 

26 

5454 
6307 

55975739588i 
6449|659i  6733 

6023 
6875 

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34 
33 

25 

26 

38i8 
4579 

3945 
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4071 

4832 

4198 

4325 

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34 

33 

27 

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32 

27 

5338 

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559i 

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5970 

32 

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8291  8432 

8574 

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28 

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6601 

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29;    8856 

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29 

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6980 

7106 

7232 

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9984  •  i25j  .265 

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29 

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7735 

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3i  9.170547 

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0828,0968 

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28 

3i 

8363 

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28 

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60"   |  50"  |  40" 

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e 

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1(V 

d 

Co-sine  of  81  Degrees. 

* 

Co-sine  of  80  Degrees. 

1 

*  Y>  «/'  nil  n»  5"  e"  7»  8»  y  [      r  j"  2"  3"  4"  3"  fi"  7"  8"  9" 

>art>  14  28  42  5fi  70  85  99  113  127  ||     ITt)  13  25  38  50  f,3  75  88  101  113 

"LOGARITHMIC    TANGENTS. 


33 


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a 

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1 

p  p  .(  1"  2''  3"  4"  5"  6"  7"  8"  9" 

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ar  \  13  26  39  52  65  78  91  103  116 

c 


LOGARITHMIC    S  i  it  K  s. 


J— 

1 

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d 

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JJ 

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3o 

9655 

9759 

9862 

9966 

..69 

.172 

29 

3i 

i3i4 

1428 

i54i!i654 

1768 

1881 

28 

3i 

9.3oo276 

o379 

0482 

0686 

o689 

0792 

32 

1994 

2IO7 

22202334 

2447 

256o 

27 

q 

0895 

0999 

1  1  02 

1206 

i3o8 

i4n 

^7 

33 

2673 

2786 

2899  3oi2 

3i25 

3238 

26 

33 

i5i4 

1617 

I72O 

1823 

I926 

2029 

26 

34 

335i 

3464 

3576  3689 

38o2 

39i5 

25 

34 

2l32 

2235 

2337 

2440 

2543 

2646 

26 

35 

4027 

4i4o 

4253^365 

4478 

459o 

24 

35 

2748 

2861 

2964 

3o57 

3i59 

3262 

24 

36 

47o3 

48i5 

4928  5o4o 

5  1  53 

5265 

23 

36 

3364 

3467 

3569 

3672 

3774 

3877 

93 

37 

5377 

549o 

56o2  57i4 

5827 

5939 

22 

37 

3979 

4082 

4i84 

4287 

4389 

449i 

22 

38 

.6o5i 

6i63 

6275  6387 

6499 

6611 

21 

38 

4593 

4696 

4798 

4900 

6002 

5io4 

21 

39 

6723 

6835 

6947  7o59 

7171 

7283 

2O 

39 

52O7 

5309 

54n 

55  1  3 

56i5 

6717 

20 

4o 

739'5 

75o6 

7618  7730 

7841 

7953 

19 

4o 

58i9 

6921 

6023 

6126 

6227 

6328 

19 

4i 

8o65 

8176 

8288,8399 

85n 

8622 

18 

4i 

643o 

6532 

6634 

6786 

6837 

6939 

18 

42 

8734 

8845 

8967  9068 

9179 

929I 

17 

42 

7o4i 

7142 

7244 

7346 

7447 

7549 

17 

43 

9402 

95i3 

96249736 

9958 

16 

43 

765o 

7762 

7853 

7955 

8066 

8168 

16 

44 

9.27oo69 

0180 

0291 

0402 

o5i3. 

0624 

i5 

44 

8259 

836o 

8462 

8563 

8664 

8766 

i5 

45 

o735 

o846 

°957 

1067 

1178 

1280. 

i4 

45 

8867 

8968 

9069 

91709272 

9373 

i4 

46 

i4oo 

i5io 

1621 

I732 

1842 

i953 

i3 

46 

9474 

9575 

9676 

97779878 

9979 

i3 

47 

2064 

2174 

2285 

2396 

25o5 

2616' 

12 

47 

9.3ioo8o 

0181 

0282 

o382 

o483 

o584 

12 

48 

2726 

2837 

2947 

3o57 

3i68 

3278 

II 

48 

o685 

0786 

0886 

o987 

1088 

1189 

II 

49 

3388 

3498 

36o8 

37i8 

3829 

3989 

10 

49 

I289 

1390 

i49o 

1691 

i692 

I792 

IO 

5o 

4049 

4i59 

4269 

4379 

4489' 

4598 

9 

5o 

i893 

i993 

2094 

219.4 

2294 

2395 

9 

5i 

4708 

48i8 

4928 

5o38 

5i48 

5267 

8 

5i 

2495 

2696 

2696 

2796 

2896 

2997 

8 

52 

5367 

5477 

5586 

5696 

58o5 

59i5 

7 

52 

3o97 

3J97 

3297 

3397 

3497 

3597 

7 

53 

6o25 

6i34 

6243 

6353 

6462 

6672 

6 

53 

3698 

3798 

3898 

3998 

4o98 

4198 

6 

54 
55 

6681 

6790 

7446 

6900 
7555 

7oo9 
7664 

7118 
7773 

7227 

7882 

5 
4 

54 
55 

4297 
4897 

4397 
4996 

4497 
6096 

46974697 
6196  6296 

4797 
5395 

5 
4 

56 

7991 

8100 

8209 

83i8 

8427 

8536 

3 

56 

5495 

5594 

56g4 

579316893 

5993 

3 

^7 

8645 

8753 

8862 

8971 

9080 

9i88 

2 

57 

6o92 

6192 

6291 

6390:6490 

6589 

2 

58 

9297 

9406 

95i4 

9623 

973i 

984o 

I 

58 

6689 

6788 

6887 

69867086 

7186 

I 

59 

9948 

..57 

.i65 

.274 

.382 

•  49i 

O 

59 

7284|7383 

7482 

76827681 

7780 

0 

60" 

50"   40" 

30" 

20" 

10" 

d 

60"     50" 

40" 

30"   20" 

10" 

d 

Co-sine  of  79  Degrees. 

§ 

Co-sine  of  78  Degrees 

£] 

,,  p   (I"  2"  3"  4"  5"  6"  7"  8"  9" 

p  p   51"  2"  3'L  4"  5"  6"  7"  8"  9" 

rt^  11  23  34  45  57  G8  80  91  102 

1  \  10  21  31"  41  52  62  72  83  93 

LOGARITHMIC    TANGENTS. 


n 

Tangent  of  10  Degrees. 

„ 

.n 

Tangent  of  1  1  Degrees. 

i 

0"     |  10" 

20" 

30"  |  40" 

50" 

^ 

0" 

10" 

20"   30" 

40"  |  50"  ) 

0 

9.246319  6442 

6565 

6688)6811 

6934|59 

o 

9.  288652 

8765 

8877(8989 

9I02  92l4J59 

I 

7°57 

7180 

73o3 

742617548 

7671 

58 

i 

9326 

9438 

9561:9663 

9775 

9887 

58 

2 

7794 

7917 

8o39 

8162 

8285 

84o7 

57 

2 

9999 

.in 

.223 

.335 

•  447 

.559 

57 

3!    853o 

8652 

8775 

8897 

902O 

9142 

56 

3 

9.290671 

o783 

oSgS 

1007 

1119 

I23l 

56 

4    9264 

9387 

95o9 

963i 

9753 

9876 

55 

4 

1  342 

i454 

i566 

1678 

I789 

1901 

55 

5 

9998 

.  120 

.242 

.364 

.486 

.608 

54 

5 

20l3 

2124 

2236 

2347 

2459 

257O 

54 

6 

9.25o73o 

o852 

o9?4 

1096 

1218 

i339 

53 

6 

2682 

279? 

2906 

3oi6 

3l27 

3239 

53 

7    i46i 

i583 

i7o5 

!826 

1948 

2070 

52 

7    335o 

346  1 

3572 

3684 

3795 

3906 

52 

8 

2191 

23i3 

2434 

2556 

2677 

2799 

5  1 

81    4017 

4128 

4239 

435i 

4462 

4573 

5i 

9 

2920 

3o4i 

3i63 

3284 

34o5 

3527 

5o 

9 

4684 

4795 

4906 

5oi6 

5l27 

5238 

5o 

10 

3648 

3769 

389o 

4on  4i32 

4253 

49 

IO 

5349 

546o 

557i 

568! 

5792 

5go3 

49 

1  1 

4374 

4495 

46i6 

4737 

4858 

4979 

48 

ii 

60  1  3 

6124 

6235 

6345 

6456 

6566 

48 

]  2 

5ioo 

5221 

534i 

5462 

5583  57o3 

47 

12 

6677 

6787 

6898 

7008 

7XI9 

7229 

47 

i3 

5824 

5945 

6o65 

6186 

63o6 

6427 

46 

i3 

745o 

756o 

7670 

7781 

7891 

46 

i4 

6547 

6668 

6788 

6908 

7029 

7149 

45 

i4 

8001 

8m 

8221 

8332 

8442 

8552 

45 

i5 

7269 

7389 

75io 

763o 

775o 

7870 

44 

i5    8662 

8772 

8882 

899.2 

9102 

9212 

44 

16 

799° 

8110 

823o 

835o 

8470 

SSgo 

43 

16 

9322 

943i 

954i 

965i 

976i 

9871 

43 

17 

6710 

883o 

8960 

9069 

9189 

93o9 

42 

17 

9980 

..90 

.200 

.309 

.419 

.529 

18 

9429 

9548 

9668 

9787 

99°7 

..27 

4i 

18 

9.3oo638 

0748 

o857 

0967 

1076 

1186 

4i 

19 

9.260146 

0266 

o385 

o5o4 

0624 

o743 

4o 

19 

1295 

i4o5 

i5i4 

1624 

I733 

1842 

4o 

20 

o863 

0982 

IIOI 

1220 

i34o 

i459 

39 

20 

1961 

2061 

2I70 

2279 

2388 

2497 

39 

21 

i578 

i697 

1816 

1935 

2o54 

2173 

38 

21 

2607 

2716 

2825 

2934 

3o43 

3i52 

38 

22 

2292 

253o 

2649 

2768 

2887 

37 

22 

3261 

337o 

3479 

3588 

3697 

38o5 

37 

23 

3oo5 

3124 

3243 

336i 

348o 

3599 

36 

23 

39i4 

4023 

4i32 

4241 

4349 

4458 

36 

24 

37i7 

3836 

3954 

4o73 

4191 

43io 

35 

24 

4567 

4675 

4784 

4893 

5ooi 

5no 

35 

25 

4428 

4547 

4665 

4783 

4902 

5O2O 

34 

25 

52i8 

5327 

5435 

5544 

5652 

576i 

34 

26 

5i38 

5256 

5375 

5493 

56n 

5729 

33 

26 

5869 

5977 

6086 

6194 

6  3  02 

64io 

33 

27 

5847 

5965 

6o83 

6201 

6319 

6437 

32 

27 

65i9 

6627 

6735 

6843 

6g5i 

7o59 

32 

r»ft 

6555 

6673 

679o 

6908 

7026 

7i44 

3i 

28 

7168 

7276 

7384 

7492 

7600 

7708 

3i 

29 

7261 

7379 

7497 

76i4 

7732 

7849 

3o 

29 

7816 

7923 

8o3i 

8i39 

8247 

8355 

3o 

3o 

7967 

8084 

8202 

8319 

8437 

85.54 

29 

3o 

8463 

857o 

8678 

8786 

8893 

9001 

29 

3i 

8671 

8789 

8906 

9023 

9140 

9258 

28 

3i 

9109 

9216 

9324 

9432 

9539 

9647 

28 

32 

9375 

9492 

9609 

9726 

9843 

996o 

27 

32 

9754 

9862 

9969 

..76 

.184 

.291 

27 

33 

9.270077 

0194 

o3n 

0428 

o545 

0662 

26 

33 

9.810399 

o5o6 

o6i3 

0720 

0828 

o935 

26 

34 

0779 

o895 

IOI2 

1129 

1246 

i362 

25 

34 

1042 

1149 

1256 

1  364 

i47i 

1578 

25 

35 

1479 

i595 

I7I2 

1829 

1945 

2062 

24 

35 

i685 

1792 

i899 

2006 

2113 

2220 

24 

36 

2178 

2294 

2411 

2527 

2644 

2760 

23 

36 

2327 

2433 

2647 

2754 

286l 

23 

37 

2876 

2992 

3  1  09 

3225 

334i 

3457 

22 

37 

2968 

3o74 

3i8i 

3288 

3394 

35oi 

22 

38 

3573 

3689 

38o5 

3921 

4o37 

4i53 

21 

38 

36o8 

37i4 

382i 

3927 

4o34 

4i4o 

21 

39 

4269 

4385 

45oi 

46i7 

4733 

4849 

2O 

39 

4247 

4353 

446o 

4566 

4672 

4779 

2O 

4o 

4964 

5o8o 

5i96 

53i2 

542  7 

5543 

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4o 

4885 

4991 

5o98 

6204 

53io 

54i6 

19 

4i 

5658 

5774 

5890 

6oo5 

6121 

6236 

18 

4i 

5523 

5629 

5735 

584i 

5947 

6o53 

18 

42 

635i 

6467 

6582 

6698 

68i3 

6928 

17 

42 

6i59 

6265 

637i 

6477 

6583 

6689 

i7 

43 

7043 

7i59 

7274 

7389 

75o4 

76i9 

16 

43 

6795 

6901 

7007 

7n3 

72l8 

7324 

16 

44 

7734 

7849 

8o79 

8194 

83o9 

i5 

44 

743o 

7536 

764i 

7747 

7853 

7958 

i5 

45 

8424 

8539 

8654 

8769 

8884 

8998 

i4 

45 

8o64 

8170 

8275 

838i 

8486 

8592 

i4 

46 

9n3 

9228 

9342 

9457 

9572 

9686 

i3 

46 

8697 

88o3 

89o8 

9oi3 

9119 

9224 

i3 

47 

98oi 

99i5 

..3o 

.i44 

'   r 

.269 

,373 

12 

47 

933o 

9435 

954o 

9645 

975i 

9856 

12 

48 

9.  280488 

0602 

o7i7 

o83i 

o945 

io59 

II 

48 

9961 

..66 

.171 

•277 

.382 

•  487 

II 

49 

1174 

1288 

1402 

i5i6 

i63o 

i744 

IO 

49 

9.320592 

0697 

0802 

o9o7 

1012 

if  17 

IO 

5o 

i858 

I973 

2087 

2201 

23i4 

2428 

9 

5o 

1222 

i326 

i43i 

i536 

i64i 

i746 

9 

5i 

2542 

2656 

2770 

2884 

2998 

3in 

g 

5i 

i85i 

i955 

2060 

2i65 

2269 

2374  8 

52 

3225 

3339 

3453 

3566 

368o 

3793 

7 

52 

2479 

2583 

2688 

2793 

2897 

30O2 

7 

53 

3907 

4021 

4i34 

4248 

436i 

4474 

6 

53 

,3io6 

3211 

33i5 

3420 

3524 

3628 

6 

54 

4588 

47oi 

48i5 

4928 

5o4i 

5i54 

5 

54 

3733 

3837 

394i 

4o46 

4i5o 

4254 

5 

55 

5268 

538i 

5494 

56o7 

5720 

5833 

•4 

55 

4358 

4463 

4567 

467i 

4775 

4879 

4 

56 

5947 

6060 

6173 

6286 

6399 

65i2 

3 

56 

4Q83 

5o87 

5i9i 

5295 

5399 

55o3 

3 

57 

6624 

6737 

685o 

6963 

7076 

7i89 

2 

b 

56o7 

5711 

58i5 

59i9 

6o23 

6127 

2 

58 

73oi 

74i4 

7527 

7639 

7752 

7865 

I 

5,8 

623i 

6334 

6438 

6542 

6646 

6749 

I 

59 

7977 

8o9o 

8202 

83i5 

8427 

854o 

O 

59 

6853 

6957 

7060 

7i64 

7267 

737ij  o 

60" 

50"   40'-' 

30" 

20" 

10" 

60"    |  50" 

40" 

30" 

20" 

10" 

j 

Co-tangent  of  79  Degrees. 

Co-tangent  of  78  Degrees. 

a 

f  I  //   Cy//   O//   All   rll   pll   nil   Qll    Q// 

P  Part  J  -1   ~   "      oo/o   y 

,  (  1"  2"  3"  4"  5"  6"  7"  8"  9" 

\  12  23  33  47  59  70  82  94  106 

P.  Part  5  n  00  30  43  54  65  ?5  86  97 

1 

36 


LOGARITHMIC   b  i  N  E  s. 


.3 

dine  of  12  Degrees. 

_c 

Sine  of  13  Degrees. 

1  " 

i 

0'    |  10" 

20"   30  "   40" 

50" 

s 

0"     10" 

20" 

30" 

40" 

50" 

0 

9.817879 

7978 

80778176 

8275 

8374 

59 

O 

9.  352088 

2I79 

2270 

2862 

2453 

2544 

59 

I 

8473 

8572 

8671  8769 

8868 

8967 

58 

I 

2635 

2726 

28l7 

2908 

2999 

8090 

58 

2 

9066 

9i65 

9263  9362 

946i 

9559 

57 

2 

3i8i 

3272 

3363 

3454 

3545 

3636 

57 

2 

9658 

9757 

98559954 

..52 

.i5i 

56 

3 

8726 

38i7 

8908 

3999 

4o9o 

4  1  80 

50 

4 

9,320249 

o348 

o446  o545 

o643 

0742 

55 

4 

4271 

4362 

4452 

4543 

4634 

4724 

55 

5 

o84o 

0938 

1087  n35 

1233 

i33c 

54 

5 

48i5 

49o6 

4996 

5o87 

5i77 

5268 

54 

6 

i43o 

i528 

1626  1724 

1822 

I92I 

53 

6 

5358 

5449 

5539 

563o 

572O 

58io 

53 

1 

2019 

2117 

22l5  23l3 

2411 

25o9 

52 

7 

59oi 

599i 

6081 

6l72 

6262 

6352 

e 

2607 

2705 

2802  2900 

2998 

3o96 

5i 

8 

6443 

6533 

6628 

67i3 

68o3 

6894 

5i 

9 

3i94 

8292 

3389|3487 

3585 

3683 

5o 

9 

6984 

7o74 

7i64 

7254 

7344 

7434 

5o 

10 

9.328780 

3878 

39754073 

4171 

4268 

49 

10 

9.357524 

7614 

7704 

7794 

7884 

7974 

49 

ii 

4366 

4463 

456i!4658 

4756 

4853 

48. 

ii 

8064 

8i54 

8243 

8333 

8428 

85i3 

48 

12 

4g5o 

5o48 

5i45 

5243 

534o 

5437 

47 

12 

86o3 

8692 

8782 

8872 

8962 

goSi 

47 

i3 

5534 

5632 

5729 

5826 

5928 

6020 

46 

i3 

9i4i 

9.281 

9320 

9410 

9499 

9589 

46 

i4 

6117 

62i5 

63i2 

64o9 

65o6 

66o3 

45 

i4 

9678 

9768 

9858 

9947 

..36 

.126 

45 

i5 

6700 

6797 

6894 

6991 

7087 

7i84 

44 

i5 

9.36o2i5 

o3o5 

o394 

o484 

o573 

0662 

44 

16 

7281 

7378 

7475 

7572 

7668 

7765 

43 

16 

0752 

o84i 

o93o 

1019 

1109 

n98 

43 

i? 

7862 

7958 

8o55 

8i52 

8248 

8345 

42 

17!    1287 

i376 

i465 

i554 

1  644 

i733 

42 

18 

844.2 

8538 

8635 

873i 

8828 

8924 

4i 

18 

.  1822 

1911 

2OOO 

2089 

2I78 

2267 

4i 

J9 

9021 

9117 

9213 

9810 

9406 

95o2 

4o 

*9 

2356 

2445 

2534 

2623 

27II 

2800 

4o 

20 

9.329599 

9695 

9791 

9888 

9984 

..80 

39 

20 

9.862889 

2978 

3o67 

3i56 

3244 

3333 

89 

21 

9.330176 

0272 

o368 

o465 

o56i 

o657 

38 

21 

3422 

35n 

3599 

3688 

3777 

3865 

38 

22 

o753 

0849 

0945 

io4i 

u37 

1233 

37 

22 

3954 

4042 

4i3i 

4220 

43o8 

4397 

37 

23 

l329 

i424 

l52O 

1616 

1712 

1808 

36 

23 

4485 

4574 

4662 

475i 

4839 

4927 

36 

24 

I9o3 

i999 

2095 

2191 

2286 

2382 

35 

24 

5oi6 

5io4 

5i93 

5281 

5369 

5457 

35 

25 

2478 

2578  2669 

2764 

2860 

2956 

34 

25 

5546 

5634 

5722 

58io 

5899 

5987 

34 

26 

3o5i 

3i473242 

3337 

3433 

3528 

33 

26 

6o75 

6168 

625i 

6339 

6427 

65i6 

33 

27 

3624 

8719 

38i4 

8910 

4oo5 

4ioo 

32 

27 

66o4 

6692 

678o 

6868 

6956 

7o44 

82 

28 

4i95 

4291 

4386 

448  1 

4576 

4671 

3  1 

28 

7181 

72I9 

73o7 

7395 

7483 

757i 

3i 

29 

4767 

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0 

60" 

50" 

40" 

30" 

20" 

10" 

a 

60" 

50" 

40" 

30"  |  20"   10" 

d 

Co-sine  cf  77  Degrees. 

i 

Co-sine  of  76  Degrees. 

X 

P  Part  5  X"  2"  3//  4"  5"  6"  7//  8"  9" 

.5  1"  2"  3"  4"  5"  6"  7"  8"  9" 

m\  9  19  28  38  47  57  GG  76  85 

)  9   18  26  35  44  53  61  70  79 

LOGARITHMIC     T  A  N  3  E  N  T  S. 


O  I 


J 

Tangent  of  12  Degrees 

d 

Tangent  of  13  Degrees. 

- 

3S 

0"    |  10" 

20"  |  30" 

40" 

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0" 

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0 

60" 

50" 

40" 

30' 

20" 

10" 

j 

60" 

50" 

40" 

30" 

20" 

10" 

Co  -tangent  of  77  Degrees. 

S 

Co-tangent  of  76  Degrees. 

S 

P  PartP"   2"   3"   4/   5"   6"   7"  8"   9// 

(  1"  2"  3"  4"  5"  6"  7"  8"  9" 

Ll£  10  20  30  43  50  60  70  80  90 

P.  Part  J  g  ig  2g  37  4(}  56  C5  ?4  Q,3 

LOGARITHMIC    SINES. 


c 

Sine  of  14  Degrees. 

•. 

a 

Sine  of  15  Degrees. 

i 

9 

0* 

10" 

20"   30"   40"   50" 

2 

0"      10" 

20" 

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50" 

0 

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27 
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2 

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9oi4 

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2O52 

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5  9 

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0 

60"    |  50"  j  40"  |  30" 

20" 

10" 

fl 

60" 

50" 

40" 

30" 

20" 

10" 

e 

Co-sine  of  75  Degrees. 

Co-sine  of  74  Degrees.    |  S 

.-,  p   (  1"  2"  5"  4"  5"  6"  7"  8"  9" 

.  <  1"  2"  3"  4"  5"  6"  7"  8''  9" 

{  8  16  24  33  41  49  57  65  73 

in\  8  15  23  30  H8  46  53  61  68 

L  O  «  A  R  IT  H  M  I  C      '1   A  N  G  E  N  T  tf. 


1 

Tangent  of  14  Degrees. 

•1 

d 

i 

Tangent  of  15  Degrees. 

0" 

10" 

20" 

30" 

40" 

50" 

0" 

W   20" 

30" 

40"   50" 

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6861 

6060 

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7i3o 

7219 

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9.  428062 

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1 

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58 

I 

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58 

2 

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1743 

1823 

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49 

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10 

49 

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52 

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7 

p* 

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3748 

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531    45o3 

4587 

4672 

4757 

4842 

4927 

6 

53 

4i48 

4228 

43o8 

4388 

4468 

4548 

6 

54 

5on 

5o96l5i8i 

6266 

535o 

5435 

5 

54 

4628 

47o8 

4787 

4867 

4947 

6027 

5 

55 

6619 

56o4 

5689 

5773 

5858 

5942 

4 

55 

5io7 

5i87 

6267 

5346 

5426 

55o6 

4 

56 

602-7 

61  12 

6r96 

6281 

6365 

645o 

3 

56 

5586 

5655 

5  745 

6826 

59o5 

5984 

3 

5? 

6534 

66i9 

67o3 

6788 

6872 

6956 

2 

57 

6064 

6i44 

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63o3 

6383 

6462 

2 

58 

7o4i 

7126 

7210 

7294 

7378 

7463 

I 

58 

6542 

6622 

6701 

6781 

6860 

694o 

I 

J>9 

7547i763r 

7716 

78oo 

7884 

7968 

0 

69 

7019 

7°99 

7!78 

7268 

7337 

74i7 

0 

W     50"   40" 

30" 

20" 

10" 

d 

60"     50" 

40"  |  30"  i  20" 

10"1  6 

Co-tangent  of  75  Degrees. 

•H 

s 

Co-tangent  of  74  Degrees.   1  M 

P  P.,  t$  *"  2//  3//  4//  5//  6"  7"  8"  9" 
11  r{  9  17  26  35  43  52  61  (19  78 

P  P-irtJ  l"  ~"  3"  4'   5"  6"  7"  8"  9</ 

11  1\  8  16  25  33  41  4^  57  65  74 

4:0 


LOGARITHMIC    SINES. 


4  I     Sine  of  16  Degrees. 

c  1     Sine  of  17  Degrees. 

I 

5 

0"    |  10' 

20"; 

30" 

40" 

50" 

sn 

•   0" 

10" 

20"  1  30" 

40''   50" 

o 

9.44o338 

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o4.85  o558 

o632 

o7o5  59 

o 

9.465935 

6oo4 

6o-3|6i42 

6211 

6280 

59 

i 

o778 

o852 

o925;o998 

1072 

n4558 

i 

6348 

64i7 

6486i6555 

6623 

6692 

58 

2 

1218 

1292 

1  365 

i438 

i5i  i 

1  584 

57 

2 

676i 

683o 

6898 

6967 

7o36 

7104 

57 

3 

i658 

i73i 

1804 

i877 

i95o 

2023 

56 

3 

7r73 

-7242 

73io 

7379 

7448 

75i6 

56 

4 

2096 

2170 

2243 

23i6 

2389 

2462 

55 

4 

7585 

7653 

7723 

779° 

7859 

7928 

55 

5 

2535 

2608 

2681 

2754 

2827 

2900 

54 

5 

7996 

8o65 

8i33 

8202 

8270 

8338 

54 

6 

2973 

3o46 

3n9 

3i92 

3265 

3337 

53 

6 

8407 

8475  8544 

8612 

8681 

8749 

53 

7 

34ro 

3483 

3556 

3629 

3702 

3774 

52 

7 

88i7 

8886 

8954 

9022 

9091 

9l59 

52 

8 

3847 

3920 

39934066 

4i38 

4211 

5i 

.8 

9227 

9296 

9364 

9432 

9569 

5i 

9 

4284 

4356 

4429*4502 

4574 

4647 

5o 

9 

9637 

97o5 

9773 

9842 

9910 

9978 

5o 

10 

9,444720 

4792 

4865  4g38 

5oio 

5o83 

49 

10 

9.  470046 

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0182 

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49 

1  1 

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5445 

55i8 

48 

ii 

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0659 

0727 

0796 

48 

12 

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5663 

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5953 

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12 

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1611 

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6748 

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lA 

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1882 

1  9  5.o 

2018 

45 

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6893 

6965 

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7io9 

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44 

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2086 

21.53 

2221 

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2357 

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44 

16 

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7398 

7470  7542 

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43 

16 

2492 

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2628 

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43 

11 

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7903:7975 

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42 

n 

2898 

2966 

3o34 

3ioi 

3i69 

3237 

42 

18 

8191 

8263 

8335 

84o7 

8479 

855i 

4i 

18 

33p4 

3372 

3439 

35o7 

3575 

3642 

4  1 

19 

8623 

8695 

8767 

8838 

89io 

8982 

4o 

19 

37ib 

3777 

3845 

3912 

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4o47 

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20 

o.449o54 

9I26 

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9269 

934i 

94i3 

39 

20 

9.474n5 

4182 

425o 

43i7 

4384 

4452 

39 

21 

9485 

9557 

9628 

9-700 

9772 

9844 

38 

21 

45i9 

4587 

4654 

47~i 

47§9 

4856 

38 

22 

99i5 

9987 

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37 

22 

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4991 

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5  1  25 

5i93 

5260 

37 

23 

9.45o345 

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36 

23 

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36 

24 

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35 

24 

573o 

5798 

5865 

5932 

5999 

6066 

35 

25 

1204 

1275 

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6268 

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6469 

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26 

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1775 

1  846 

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26 

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27 

2060 

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2203  2274 

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32 

27 

6938 

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7072 

7l39 

7206 

727332 

28 

2488 

2559 

263o  2702 

2773 

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28 

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74o7 

7473 

754o 

7607 

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29 

2915 

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320O 

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7875 

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9.453342 

34i3 

3484 

3555 

3626 

3697 

29 

3o 

8209 

8275 

8342 

84o9 

8476 

29 

3i 

3768 

3839 

39io398i 

4o52 

4l23 

28 

3i 

8542 

8609 

8676 

8742 

8809 

8876 

28 

32 

4194 

4^65 

43364407 

4477 

4548 

27 

32 

8942 

9009 

9076 

9142 

9209 

9275 

27 

33 

4619 

469o 

4761 

4832 

4903 

4973 

26 

33 

9342 

9409 

9475 

9542 

9608 

9675 

26 

34 

5o44 

5n5 

5i86 

5a56 

5327 

5398 

25 

34 

o?4i 

0808 

0874 

of  \A  i 

74 

25 

35 

5469 

5539 

56io568i 

/ 

575i 

7 

5822 

24 

35 

V  t**T 

9.48oi4o 

V 

O207 

v  w  /•* 
0273 

oJ39 

0406 

0472 

24 

36 

5893 

5963 

6o34 

6io4 

6i75 

6246 

23 

36 

o539 

o6o5 

0671 

o738 

0804 

0870 

23 

37 

63i6 

6387 

6457 

6528 

6598 

6669 

22 

37 

o937 

ioo3 

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1202 

1268 

22 

38    6739 

6810 

68806951 

7O2I 

709J 

21 

38 

1  3  34 

i4oo 

i467 

i533 

i599 

i665 

21 

39 

7162 

7232 

73o3 

7373 

7443 

75i4 

2O 

39 

I73i 

i798 

1  864 

1930 

i996 

2062 

2O 

4o 

9.4'57584 

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7725 

7795 

7865 

7936 

19 

4o 

9.482128 

2I94 

2261 

2327 

2393 

2459 

19 

4i 

8006 

8076 

8i46 

82I7 

8287 

8357 

18 

4i 

2525 

259I 

2657 

2723 

2789 

2855 

18 

42 

8427 

8497 

8567 

8638 

87o8 

8778 

i7 

42 

2921 

2987 

3o53 

3119 

3i85 

325i 

17 

43 

8848 

8918 

89S8!9o58 

9128 

9198 

16 

43 

33i6 

3382 

3448 

35i4 

358o 

3646 

16 

44 

9268 

9338 

94o8:9478 

9548 

9618 

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44 

37I2 

3778 

3843 

3909 

3975 

4o4i 

i5 

45 

9688 

9758 

9828 

9898 

9968 

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i4 

45 

4io7 

4l72 

4238 

43o4 

4370 

4435 

i4 

46  9.460108 

0178 

0248 

o3i7 

o387 

o457 

i3 

46 

45oi 

4567 

4632 

4698 

4764 

4829 

i3 

47I    o527 

o597 

o667 

o736 

0806 

o876 

12 

47 

4895 

4961 

0026 

5092 

5i58 

D223 

12 

48 

o946 

ioi5 

io85 

n55 

1224 

1294 

II 

48 

5289 

5354 

5420 

5485 

555i 

56i7 

II 

49    1364 

i433 

i5o3 

i573 

1642 

I7I2 

10 

49 

5682 

5748 

58i3 

5879 

5944 

6009 

IO 

509.461782 

i85i 

I92I 

i99o 

2060 

2129 

9 

5o 

9.486o75 

6i4o 

6206 

6271 

6337 

6402 

9 

5i 

2I99 

2268 

2338 

2477 

2546 

8 

5i 

6467 

6533 

6598 

6664 

6729 

6794 

8 

52 

2616 

2685 

2755 

2824 

2894 

2963 

7 

52 

6860 

6925 

6990 

7o55 

7121 

7186 

7 

53 

3o32 

3l02 

3i7i 

324o 

33io 

3379 

6 

53 

725l 

73i6 

7382 

7447 

7512 

7577 

6 

54 

3448 

35i8 

3587 

3656 

3725 

3795 

5 

54 

7643 

7708 

7773 

7838 

79°3 

7968 

5 

55 

3864 

3933 

40O2 

4o72 

4i4i 

4210 

4 

55 

8o34 

8099 

8i64 

8229 

8294 

8359 

4 

56 

4279 

4348 

44i7 

4486 

4556 

4625 

3 

56 

8424 

8489 

8554 

8619 

8684 

8749 

3 

57 

4694 

4763 

4832 

49oi 

497o 

5o39 

2 

57 

88i4 

8879 

8944 

9009 

9074 

9l39 

9 

5S 

5io8 

5i77 

5246 

53i5 

5384 

5453 

I 

58 

9204 

9269 

9334 

9399 

9464 

9528 

I 

J? 

5522 

559i 

566o 

57295798 

5866 

o 

59 

9593 

9658 

9723 

9788 

9853 

9918 

0 

6-y- 

50« 

40" 

30"  |  20' 

10" 

c 

60"    |  50" 

40" 

30"  |  20"  j  10" 

. 

Co-sine  of  73 

Degrees. 

Co-sine  of  72  Degrees. 

1 

p.  Part  |  y  *'  *'j  048' 

36  43  50  57  64 

i 

i  I//  o"  Q"   A.11   <^H   I''1  7"   C"   Q>; 
P  Part  Jl~d45(j7 

irt}  7  13  20  27  33  40  47  53  60 

LOGARITHMIC    TANGENTS. 


fil 

Tangent  of  16  Degrees. 

1 

Tangent  of  17  Degrees. 

- 

* 

0" 

10" 

20"  |  30" 

40" 

50" 

's 

0" 

10" 

20"   30" 

40" 

50' 

o 

9.457496 
7973 

7576 
8o52 

7655]7735 
8l3282II 

78i4 
829o 

7894 

837o 

59 

58 

0 

I 

9.485339 

579i 

54i4 
5866 

5490 

594i 

5565 
6016 

564o 
6092 

57i5 
6i67 

59 

58 

a 

8449 

8528 

8608 

8687 

8766 

8846 

57 

2 

6242 

63I7 

6392 

6467 

6543 

6618 

57 

3 

8925 

900^ 

9083 

9i63 

9242 

9321 

56 

3 

6693 

6768 

6843 

6918 

6993 

•7068 

56 

4 

9400 

9479 

9558 

9638 

97i7 

9796 

55 

4 

7i43 

72l8 

7293 

7368 

7443 

75i8 

55 

5 

9875 

9954 

..33 

.  112 

.191 

.270 

54 

5 

7593 

7668 

7743 

78i8 

7893 

7968 

54 

6 

9.460349 

0428 

o5o7 

o586 

o665 

0744 

53 

6 

8o43 

8118 

8i93 

8268 

8343 

84i8 

53 

7 

0823 

0902 

0981 

1060 

1139 

1218 

52 

7 

8492 

8567 

8642 

8717 

8792 

8866 

52 

8 

1297 

i376 

i454 

i533 

1612 

1691 

5i 

8 

894i 

9016 

9091 

9166 

9240 

93i5 

5i 

9 

1770 

1849 

I927 

2006 

2o85 

2164 

5o 

9 

939o 

9465 

9539 

9614 

9689 

9763 

5o 

10 

9.462242 

2321 

240O 

2478 

2557 

2636 

49 

10 

9.489838 

99i3 

9987 

..62 

.i37 

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49 

ii 

2715 

2793 

2872 

2950 

3029 

3io8 

48 

ii 

9.49Q286 

o36o 

o435 

o5io 

o584 

o659 

48 

12 

3:86 

3265 

3343 

3422 

35oi 

3*79 

47 

12 

o733 

0808 

0882 

o957 

io3i 

1  1  06 

47 

i3 

3658 

3736 

38i5 

3893 

3972 

4o5o 

46 

i3 

1180 

1255 

1329 

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i478 

i55a 

46 

i4 

4128 

4207 

4285 

4364 

444s 

4521 

45 

i4 

1627 

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i776 

i85o 

1924 

1999 

45 

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4599 

4677 

4756 

4834 

4912 

4991 

44 

i5 

2073 

2l47 

2222 

2296 

2370 

2445 

44 

16 

5o69 

5i47 

5226 

53o4 

5382 

54Po 

43 

16 

25i9 

2593 

2668 

2742 

2816 

2890 

43 

17 

5539 

56i7 

5695 

5773 

585i 

5g3o 

42 

J7 

2965 

SoSg 

3n3 

3i87 

3261 

3336 

42 

18 

6008 

6086 

6164 

6242 

6320 

6398 

4i 

18 

34io 

3484 

3558 

3632 

37o6 

378o 

4i 

'9 

6477 

6555 

6633 

6711 

6789 

6867 

4o 

J9 

3854 

3929 

4oo3 

4o77 

4i5i 

4225 

4o 

20 

9.466945 

7023 

7101 

7179 

7257 

7335 

39 

20 

9-494299 

4373 

4447 

452i 

4595 

4669 

o 

21 

74i3 

7491 

7569 

7647 

7724 

7802 

38 

21 

4743 

48i7 

489i 

4965 

SoSg 

5n3 

38 

22 

7880 

7958 

8o36 

8n4 

8192 

8269 

37 

22 

5i86 

5260 

5334 

54o8 

5482 

5556 

37 

23 

8347 

8425 

85o3 

858: 

8658 

8736 

36 

23 

563o 

57o4 

5777 

585i 

5925 

5999 

36 

Si 

8814 

8892 

8969 

9047 

9125 

9202 

35 

24 

6o73 

6i46 

6220 

6294 

6368 

644  1 

35 

25 

9280 

9358 

9435 

95i3 

959i 

9668 

•M 

25 

65i5 

6589 

6663 

6736 

6810 

6884 

34 

26 

9746 

9823 

9901 

9979 

..56 

.i34 

o3 

26 

6957 

7o3i 

7io5 

7i78 

7252 

7326 

33 

27 

9.470211 

0289 

o366 

o444 

0521 

o599 

32 

27 

7399 

7473 

7546 

7620 

7693 

7767 

32 

28 

0676 

o754 

o83i 

0909 

o986 

io63 

3i 

28 

784i 

-7914 

-7988 

8061 

8i35 

8208 

3i 

29 

i  i4i 

1218 

1295 

i373 

i45o 

i528 

3o 

29 

8282 

8355 

8429 

85o2 

8575 

8649 

3o 

3o 

9.471605 

1682 

1759 

i837 

i9i4 

1991 

29 

3o 

9.498722 

8796 

8869 

8943 

9016 

9089 

29 

Si 

2069 

2146 

2223 

2300 

2377 

2455 

28 

3i 

.  9i63 

9236 

9309 

9383 

9456 

9529 

28 

32 

2532 

2609 

2686 

2763 

2840 

2910 

27 

32 

96o3 

9676 

9749 

9822 

9896 

9969 

27 

33!    2996 
34J    3457 

3072 
3534 

3i49 
36n 

3226 

3688 

33o3 
3765 

338o 

3842 

26 
25 

.S3 

34 

9.5ooo42ioii5 
0481  o555 

0189 
0628 

0262 

O7OI 

o335 

o774 

o4o8 
o847 

26 

25 

35    3919 

3996 

4o73 

4i5o 

4227 

43o4 

24 

35 

092O 

0993 

1066 

n4o 

I2l3 

1286 

24 

36 

438  1 

4458 

4535 

46i2 

4688 

4765 

23 

36 

i359 

1432 

i5o5 

i578 

i65i 

I724 

23 

3? 

4842 

4919 

4996 

5o73 

5i49 

5226 

22 

37 

i797 

1870 

1943 

2016 

2089 

2162 

22 

38 

53o3 

538o 

5456 

5533 

56io 

5687 

21 

38 

2235 

23o8 

238i 

2453 

2526 

2599 

21 

39 

5763 

584o 

5917 

5993 

6070 

6i47 

2O 

39 

2672 

2745 

2818 

280.1 

2964 

3o36 

2O 

4o 

9.476223 

63oo 

6377 

6453 

653o 

6606 

*9 

4o 

9.5o3iog 

3i82 

3255 

3328 

34oo 

3473 

ig 

4i 

6683 

6759 

6836 

6913 

6989 

7o66 

18 

4i 

3546 

3619 

3691 

3764 

3837 

3910 

18 

42 

7142 

7219 

7295 

7372 

7448 

7524 

17 

42 

3982 

4o55 

4128 

4200 

4273 

4346 

ll 

43 

7601 

7677 

7754 

783o 

7906 

7983 

16 

43 

44i8 

449i 

4563 

4636 

47Q9 

478i 

16 

44 

8059 

8i36 

8212 

8288 

8365 

844i 

i5 

44 

4854 

4926 

4999 

5o72 

5i44 

52J7 

i5 

45 

85i7 

8593 

8670 

8746 

8822 

8899 

i4 

45 

6289 

5362 

5434 

55o7 

5579 

5652 

i4 

46 

8975 

9o5i 

9127 

9203 

9280 

9356 

i3 

46 

5724 

5796 

5869 

594i 

6oi4 

6086 

i3 

47 

9432 

95o8 

9584 

9660 

9737 

98i3 

12 

47 

6159 

623i 

63o3 

6376 

6448 

652O 

12 

48 

9889 

9965 

..4i 

.n7 

.i93 

.269 

II 

48 

6593 

6665 

6737 

6810 

6882 

6954 

II 

49 

9.48o345 

0421 

0497 

o573 

0649 

O725 

IO 

49 

7027 

7099 

7171 

7243 

73i6 

7388 

10 

5o 

9.480801 

0877 

o953 

1029 

i  io5 

1181 

9 

5o 

g.5o746o 

7532 

76o5 

7677 

7749 

7821 

9 

5: 

1257 

1333 

i4o8 

i484 

i56o 

i636 

8 

5i 

7893 

7965 

8o38 

8110 

8182 

8254 

8 

52 

1712 

1788 

i863 

i939 

20l5 

209I 

7 

52 

8326 

8398 

847o 

8542 

86i4 

8687 

7 

53 

2167 

2242 

23i8 

2394 

2470 

2545 

6 

53 

8759 

883i 

89o3 

8975 

9o47 

9:JI9 

6 

54 

2621 

2697 

2772 

2848 

2924 

2999 

5 

54 

9i9i 

9263 

9335 

94oy 

9479 

955i 

5 

55 

3075 

3i5i 

3226 

33o2 

3377 

3453 

4 

55 

9622 

9694 

9766 

9838 

9910 

9982 

4 

55 

3529 

36o4 

368o 

3755 

383i 

39o6 

3 

56 

y.5ioo54 

0126 

0198 

0269 

o34i 

o4i3 

3 

57 

3982 

4o57 

4i33 

4208 

4284 

4359 

2 

57 

o485 

o557 

0629 

o7oo 

0772 

o844 

2 

58 

4435 

45io 

4585 

466i 

4736 

4812 

I 

58 

o9i6 

o987 

loSg 

n3i 

1203 

1274 

I 

59 

4887 

4962 

5o38 

5n3 

5i88 

5264 

O 

59 

1  346 

i4i8 

1489 

i56i 

i633 

i7o4 

0 

60" 

50" 

40" 

30" 

20" 

10" 

£ 

(0" 

50" 

40" 

30" 

20" 

10" 

JH 

Co-tangent  of  73  Degrees. 

i 

Co-tangent  of  72  Degrees. 

9 

p  p   <  I"  2"  3"  4"  5"  6"  7"  8"  9" 

p  P  t  $  1"  2"  3"  4"  5"  6"  7"  8"  9" 

>  8  15  23  31  39  46  54  62  70 
i  

I  7   15  22  29  37  44  51  59  66 

LOGARITHMIC   SINE?. 


17 

Sine  of  18  Degrees. 

B 

Sine  of  19  Degrees. 

m 

0" 

10" 

20"   30"   40" 

50" 

i 

0"      10" 

20" 

30" 

40" 

50" 

0 

9.489982 

..47 

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f 

o 

9.  512642 

27o3 

2764 

2825 

2886 

2948 

59 

i 

9.49o37i 

o436 

o5oo  o565  o63o 

o695 

58 

I 

3oo9 

3070 

3i3i 

3i92 

3253 

33i4 

58 

2. 

0759 

0824 

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1618 

1082 

57 

? 

3375 

3436 

3497 

3558 

36i9 

368o 

57 

3 

n47 

1212 

1276  i34i 

i4o5 

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56 

3 

.   374i 

38o2 

3863 

3924 

3985 

4o46 

56 

A 

i535 

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1664  1728 

i793 

i857 

55 

l\ 

4107 

4i68 

4229 

4289 

435o 

44n 

55 

5!      I922 

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224^ 

54 

5 

4472 

4533 

4594 

4655 

47i5 

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54 

6 

23o8 

2373 

2437  2502  2566 

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53 

6 

4837 

4898 

4959 

5oi9 

5o8o 

5i4i 

53 

7 

2695 

2759 

2823288820.52 

3oi6 

52 

7 

52O2 

5262 

5323 

5384 

5445 

55o5 

52 

8 

3o8i 

3i45 

3209  3273 

3338 

3402 

5i 

5566 

5627 

5687 

574S 

58o9 

5869 

5i 

9 

3466 

353o 

3595  3659 

3723 

3787 

5o 

9 

593o 

599i 

6o5i 

6112 

6172 

6233 

5o 

10 

9.49385i 

39i5 

398o  4o44 

4io8 

49 

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9  5i629^ 

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64i5 

6475 

6536 

6596 

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ii 

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43oo  4364  4428 

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6717 

6778 

6838 

6899 

6959 

48 

12 

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494i 

47 

12 

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7080 

7i4i 

7201 

7262 

7322 

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5324 

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7382 

7443 

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7624 

7684 

46 

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5388 

5452 

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45 

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6o27 

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44 

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8io7 

8167 

8227 

8287 

8348 

84o8 

44 

16 

6i54 

6218 

6282  6346 

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6473 

43 

16 

8468 

8528 

8589 

8649 

87o9 

8769 

43 

J7 

6537 

6601 

66646728 

6792 

6856 

42 

17 

8829 

889o 

895o 

9oio 

9o7o 

9i3o 

42 

18 

69i9 

6983 

7047  7110 

7i74 

7237 

4i 

18 

9i9o 

925o 

93ii 

937i 

943i 

949i 

4i 

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73oi 

7365 

74?8  749? 

7555 

76i9 

4o 

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955i 

96ii 

967i 

973i 

979i 

985i 

4o 

20 

9.497682 

7746 

^7810  7873 

7937 

8000 

39 

20 

9.519911 

9971 

.  .3i 

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39 

21 

8o64 

8127 

810,08254 

83i7 

838i 

38 

21 

9.52O27I 

o33i 

o39i 

o45  1 

o5n 

o57i 

38 

22 

8444 

85o8857i  8634 

8698 

876i 

37 

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60"    j  50" 

40" 

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. 

60" 

50"   40" 

30" 

20" 

i)" 

c 

Co-sine  of  71  Degrees. 

S3 

Co-sine  of  70  Degrees. 

i 

P  Part  5  L//  2"  3"  4"  5"  6"  7"  8"  9"  \ 

.  (  1"  2"  3"  4"  5"  6"  7"  8"  9" 

\  G   13  19  25  31  38  44  50  57 

"j  6  1-2  18  24  30  36  42  48  54 

LOGARITHMIC    TANGEK'TS. 


it  O 



s     Tangent  of  18  Degrees. 

a 

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60"  •   |  50" 

40" 

30" 

20" 

10" 

S* 

60"  '   |  50" 

40" 

30" 

20" 

10" 

a 

Co-tangent  of  7  1  Decrees. 

Co-tangent  of  70  Degrees. 

i 

p  p   (  I"  2"  3"  4"  5"  6"  7"  8"  9" 

(1"  2"  3"  4"  5"  6"  7"  8"  9" 

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l<  7  13  20  27  33  4°  47  54  G0 

lj  O  G  A  R  I  T  II  M  1  C     IS  I  N  E  S. 


2  |     Sine  of  20  Degrees. 

d 

Sine  of  21  Degrees. 

- 

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10" 

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0" 

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w 

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3286 

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2684 

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3396 

345i 

35o6  356o 

36i5 

2 

57 

2636 

2688 

274l 

2793 

2845 

2897 

2 

58 

36  70 

3725 

378o 

3835 

3890 

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2960 

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4i  10 

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3263 

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3523 

0 

60'     50" 

40" 

30"  |  20"  |  10" 

a 

60" 

50"   40" 

30'   20" 

10" 

d 

Co-sine  of  69  Degrees. 

2 

Co-sine  of  68  Degrees.  - 

a 

.<  1"  2"  3"  4"  5"  G"  7"  8"  9" 

p  p  .  (  1"  2"  3"  4"  5"  6"  7  '  8"  9' 

1  \  6  11  17  93  23  34  39  45  51 

1  I  5  11  16  21  27  32  37  43  48 

LOGARITHMIC    TANGENTS. 


1 

Tangent  of  20  Degrees. 

50"  j 

j 

Tangent  of  21  Degrees. 

1 

0"      10" 

20" 

30" 

40" 

S     0" 

10"  |  20" 

30" 

40" 

50" 

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l/.  661666 

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1917 

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3 

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4 

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5 

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6 

34  1  9 

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52 

7 

68i5 

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6940 

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52 

8 

4202 

4267 

4332 

4397 

4462 

4527 

5i 

8 

7190 

7253 

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7378 

744  1 

75o3 

5r 

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4658 

4723 

4788 

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49 

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3432 

8 

52 

1149 

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1402 

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7 

52 

3493 

3554 

36i5 

3675 

3736 

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7 

53 

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1718 

1781 

1  844 

6 

53 

3858 

39i9 

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4o4i 

4102 

4162 

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54 

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1971 

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5 

54 

4223 

4284 

4345 

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4527 

5 

55 

2286 

235o 

24i3 

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2539 

2602 

4 

55 

4588 

4649 

4710 

477i 

483i 

4892 

4 

56 

2665 

2728 

2791 

2854 

2917 

298o 

3 

56 

4953 

5oi4 

5o74 

5i35 

5196 

5257 

3 

57 

3o44 

3107 

3170 

3233 

3296 

3359 

2 

57 

53i7 

5378 

5439 

55oo 

556o 

562i 

2 

58 

3422 

3485 

3548 

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3674 

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58 

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59 

38oo 

3863 

3926 

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59 

6o46j6io6 

6167 

6228 

6288 

6349 

C 

60" 

50"  |  40" 

30" 

20" 

10" 

q 

60"    |  50"   40" 

30" 

20" 

10'' 

d 

Co-tangent  of  69  Degrees. 

i 

Co-tangent  of  68  Degrees. 

i 

»  Ps,rf$  1"  2"  3"  4"  5"  6"  7"  8"  9" 

p  T>  .  $  I"  2"  3"  4"  5"  6"  7"  8"  9" 

£  6  13  19  26  32  39  45  51  58 

\  6   12  If)  25  31  37  43  49  5G 

LOGARITHMIC    SINES. 


i  .5 

Sine  of  22  Degrees. 

d 

Sine  of  23  Degrees. 

2 

0' 

10" 

20"   30"   40" 

50" 

s 

0" 

10" 

20" 

30"  |  40" 

50" 

0 

9.5735^5 

3628 

368o3732  3784 

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0 

9.59i878 

1928 

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2027 

2076 

2120 

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3888 

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58 

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56 

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2819 

2869 

2918 

2968 

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4 

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5o84 

55 

4 

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32i5 

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55 

c 

5i36 

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5343 

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54 

c 

3363 

34i2 

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6 

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53 

6 

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7 

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52 

7 

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6276 

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4399 

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9 

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6482  6534 

6586 

6638 

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9 

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9.576689 
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9213 

92649316 

9367 

94i8 

4i 

18 

•7196 

7245 

7294 

7343 

7392 

744i 

4i 

'9 

9470 

9521 

9572 

9623 

9675 

9726 

4o 

J9 

749o 

7539 

7587 

7636 

7685 

7734 

4o 

20 

9.579777 

9828988o993i 

9982 

..33 

39 

20 

9.597783 

783i 

788o 

7929 

7978 

8o27 

39 

21 

9.58oo85 

oi36 

01870238 

0289 

o34i 

38 

21 

8o75 

8124 

8i73 

8222 

82-70 

83i9 

38 

22 

0392 

o443 

o4g4 

o545 

0596 

o647 

37 

22 

8368 

84i7 

8465 

85i4 

8563 

8612  37 

23 

0699 

o75o  0801 

o852 

0903 

o954 

36 

23 

8660 

87o9 

8758 

8806 

8855 

8904 

36 

24 

1006 

io56 

1107 

u58 

1209 

1261 

35 

24 

8952 

9ooi 

9o5o 

9098 

9i47 

9i95 

35 

25 

l3l2 

i363 

1414 

i465 

i5i6 

1567 

34 

25 

9244 

9293 

934i 

939o 

9438 

9487 

34 

26 

1618 

i669 

1720  1771 

1822 

i873 

33 

26 

9536 

9584 

9633 

9681 

973o 

9778 

33 

27 

1924 

1975 

2O25  2076 

2127 

2178 

32 

27 

982y 

9876 

9924 

9973 

.  .21 

..70 

32 

28 

2229 

2280 

233i  2382 

2433 

2484 

3i 

28 

9.  600118 

oi67 

0215 

0264 

O3l2 

o36i 

3i  i 

29 

2535 

2585 

2636 

2687 

2738 

2789 

3o 

29 

o4o9 

o457 

o5o6 

o554 

o6o3 

o65r 

3o 

3o 

9.582840 

2890 

2941 

2992 

3o43 

3o94 

29 

3o 

9.  600700 

o748 

°797 

o845 

o893 

o942 

29 

3i 

3i45 

3195 

3246 

3297 

3348 

3398 

28 

3i 

o99o 

io38 

io87 

n35 

n84 

1232 

28 

32 

3449 

35oo 

355i 

36oi 

3652 

37o3 

27 

32 

1280 

i329 

i377 

i425 

1474 

1522 

27 

33 

3754 

38o4 

3855 

39o6 

3956 

4007 

26 

33 

i57o 

i6i9 

i667 

i7i5 

i763 

1812 

26 

34 

4o58 

4i68  4i59 

4210 

4260 

43u 

25 

34 

1860 

I9o8 

i957 

2005 

2o53 

2IOI 

25 

35 

436i 

44i24463 

45i3 

4564 

46i5 

24 

35 

2i5o 

2I98 

2246 

2294 

2342 

239I 

24 

36 

4665 

47164766 

48i7 

4867 

49i8 

23 

36 

a439 

2487 

2535 

2583 

2632 

2680 

23 

37 

4968 

6019  5o7o 

5l20 

5i7i 

5221 

22 

37 

2728 

2776 

2824 

2872 

292O 

2969 

22 

38 

5272 

5322 

5373 

5423 

5474 

5524 

O  T 

38 

3oi7 

3o65 

3n3 

3i6i 

32O9 

3257 

21 

39 

4o 

5574 
9.  585877 

5625 
5927 

5675 
5978 

5726 
6028 

5  7  76 
6o79 

5827 

6l29 

2O 
19 

39 
4o 

33o5  "353  34oi 
9.6o3594  3642  "69o 

3449 
3738 

3497 
3786 

3546 
3834 

20 
19 

4i 

6179 

6230^280 

633i 

638i 

643i 

18 

4i 

3882  393o 

3978 

4026 

4074 

4122 

18 

42 

6482 

6532 

6582 

6633 

6683 

6733 

*7 

42 

4i7o 

4218 

4266 

43i3 

436i 

44o9 

n 

43 

6783 

6834 

6884 

6934 

6985 

7o35 

16 

43 

4457 

45o5 

4553 

46oi 

4649 

4697 

16 

44 

7o85 

7I35 

7i86 

7236 

-7286 

7336 

i5 

44 

4745 

4793 

484i 

4888 

4936 

4984 

i5 

45 

7386 

7437 

7487 

7537 

7587 

7637 

i4 

45 

5o32 

5o8o 

5i28 

5176 

5223 

527I 

i4 

46 

7688 

7738 

7788 

7838 

•7888 

7938 

i3 

46 

53i9 

5367 

54i5 

5462 

55io 

5558 

i3 

47 

7989 

80398089 

8i39 

81898239 

12 

47 

56o6 

5654 

57oi 

5749 

5797 

5845 

12 

48 

8289 

83398389 

8439 

8489  854o 

II 

48 

5892 

594o 

5988 

6o35 

6o83 

6i3i 

II 

49 

859o 

864o  8690 

874o 

879o  884o 

IO 

49 

61-79 

6226 

6274 

6322 

6369 

64i7 

IO 

5o 

9.588890 

8g4o  8990 

9o4o 

9090  9140 

9 

5o 

9.6o6465 

65i2 

656o 

6608 

6655 

67o3 

9 

5i 

9190 

9240  9290 

934o 

9389  9439 

8 

5i 

675i 

6798 

6846 

6893 

6941 

6989 

8 

52 

9489 

95399589 

9639 

9689  9739 

7 

52 

7o36 

7o84 

7i3i 

7179 

7227 

7274 

7 

53 

9789 

98399889 

9938 

9988..  38 

6 

53 

7322 

7369 

74i7 

7464 

7512 

7559 

6 

54 

9.590088 

oi38oi88 

0237 

O287o337 

5 

54 

7607 

7654 

7702 

7749 

7797 

7844 

5 

55 

o387 

o437  o487 

o536 

o586o636 

4 

55 

7892 

7939 

7987 

8o34 

8082 

8i29 

4 

56 

0686 

o735 

o785 

o835 

o885  o934 

3 

56 

8177 

8224 

827I 

83i9 

8366 

84U 

3 

57 

0984 

io34 

1084 

n33 

u83  1233 

2 

57 

846  1 

85o8 

8556 

86o3 

865i 

8698 

2 

58 

1282 

i332 

i382 

i43i 

i48i  i53i 

I 

58 

8745 

8793 

884o 

8887 

8935 

8982 

I 

59 

i58o 

i63o 

1680 

1729 

i779  1828 

O 

59 

9029 

9°77 

9124 

9171 

9219 

9266 

O 

60" 

50" 

40"  |  30" 

20"   10" 

A 

60" 

50"  |  40"  |  30" 

20" 

10" 

d 

Co-sine  of  67  Degrees. 

.3 
& 

Co-sine  of  66  Degrees.     § 

P  Part*  l"  ~"  3/ 

'  4"  5"  6"  7"  8"  9" 

.  (  1"  2"  3"  4"  5"  6"  7"  8"  9"  i 

irt{  5  10  1; 

»  20  25  31  36  41  46 

irt}  5  10  15  19  24  29  34  39  44  j 

LOGARITHMIC    TANGENTS. 


A 

Tangent  of  22  Degrees. 

a 

Tangent  of  23  Degrees. 

Si 

0'' 

10" 

20" 

30" 

40"  |  50" 

JJ 

0" 

10"   20" 

30"   40"   50" 

o 

9.6o64io 

6470 

653i 

659i 

6652 

67i3 

59 

O 

9.627852 

79io 

7969 

8028 

8086 

8i45 

59 

j 

6773 

6834 

6894 

6955 

7oi5 

7076 

58 

I 

82o3 

8262 

8320 

8379 

8437 

8496 

58 

2 

7i37 

7197 

7258 

73i8 

7379 

7439 

57 

2 

8554 

8612 

867i 

8729 

8788 

8846 

57 

3 

75oo 

756o 

762I 

7681 

7742 

7802 

56 

3 

8905 

8963 

9O22 

9080 

9i38 

9197 

56 

4 

7863 

7923 

7984 

8o44 

8io5 

8i65 

55 

4 

9255 

93i4 

9372 

943i 

9489 

9547 

55 

5 

8225 

8286 

8346 

84o7 

S467 

8528 

54 

5 

9606 

9664 

9722 

978i 

9839 

9897 

54 

6 

8588 

8648 

87o9 

8769 

883o 

S89o 

53 

6 

9966 

.  .14 

..73 

.i3i 

.189 

.247 

53 

7 

895o 

9011 

90-71 

9i3i 

9192 

92D2 

62 

7 

9.63o3o6 

o364 

O422 

o48i 

o539 

o597 

5s 

8 

93l2 

9373 

9433 

9493 

9554 

96i4 

5i 

8 

o656 

07  1  4 

0772 

o83o 

o889 

0947 

5i 

9 

9674 

9735 

9795 

9855 

9915 

9976 

5o 

9 

ioo5 

io63 

1122 

1180 

1238 

I296 

5o 

10 

9.6ioo36 

0096 

oi56 

0217 

0277 

o337 

49 

ro 

9.  63j-355 

i4i3 

l47I 

1529 

i587 

1  646 

49  1 

ii 

o397 

o458 

o5i8 

o578 

o638 

0698 

48 

ii 

-  1704 

1-762 

1820 

i878 

936 

1995 

48 

12 

o759 

0819 

o879 

o939 

°999 

io59 

47 

12 

2o53 

2III 

2l69 

2227 

2385 

2343 

47 

i3 

1120 

1180 

1240 

i3oo 

i36o 

l42O 

46 

i3 

2402 

246o 

25i8 

2576 

2634 

2692 

46 

i4 

i48o 

i54o 

1601 

1661 

1721 

1781 

45 

1  4' 

2750 

2808 

2866 

2924 

2982 

3o4o 

45 

i5 

i84i 

1901 

i96i 

2021 

2081 

2l4l 

44 

i5 

3o99 

3  1  57 

32i5 

3273 

33  3  1 

3389 

44 

16 

22OI 

2261 

2321 

238i 

244i 

2501 

43 

16 

3447 

35o5 

3563 

362i 

3679 

3737 

43 

*7 

256l 

2621 

2681 

274l 

2801 

286l 

42 

17 

3795 

38.5.3 

39n 

3969 

4027 

4o85 

42 

18 

292I 

2981 

3o4i 

3ioi 

3i6i 

3221 

4i 

18 

4i43 

4201 

4259 

43i6 

4374 

4432 

4r 

19 

328l 

334i 

34oi 

346i 

352i 

358i 

4o 

19 

449o 

4548 

46o6 

4664 

4722 

478o 

4o 

20 

9.6i364i 

37oi 

376o 

3820 

388o 

394o 

39 

20 

9.  634838 

4896 

4954 

Son 

5o69 

5l27 

39 

21 

4ooo 

4o6o 

4l20 

4i8o 

4239 

4299 

38 

21 

5i85 

5243 

53oi 

5359 

54i6 

5474 

38 

22 

4359 

44i9 

4479 

4539 

4598 

4658 

37 

22 

5532 

559o 

5648 

57o6 

5763 

582i 

37 

23 

47i8 

4778 

4838 

4897 

4957 

5017 

36 

23 

5879 

5937 

5995 

6o52 

61  10 

6168 

36 

24 

5o77 

5i36 

5i96 

5256 

53i6 

5375 

35 

24 

6226 

6283 

634i 

6399 

6457 

65i4 

35 

25 

5435 

5495 

5555 

56i4 

5674 

5734 

34 

25 

6572 

663o 

6688 

6745 

68o3 

6861 

34 

26 

5793 

5853 

59i3 

5972 

6o32 

6092 

33 

26 

6919 

6976 

7034 

7092 

7i49 

7207 

33 

27 

6i5i 

6211 

627I 

633o 

639o 

645o 

32 

27 

7265 

7322 

738o 

7438 

7495 

7553 

32 

n 

65o9 

6569 

6628 

6688 

6748 

6807 

3  1 

28 

76n 

7668 

7726 

7783 

7841 

7899*31 

29 

6867 

6926 

6986 

7046 

7io5 

7i65 

3o 

29 

•7956 

8oi4 

80-72 

8i29 

8i87 

8244 

3o 

3o 

^  .  6f7224 

7284 

7343 

74o3 

7462 

7522 

29 

3o 

9.  638302 

8359 

84i7 

8475 

8532 

859o 

29 

3i 

7582 

764i 

77oi 

7760 

7820 

7879 

28 

3i 

8647 

87o5 

8762 

8820 

8877 

8935 

28 

32 

7939 

7998 

8o57 

8117 

8176 

8236 

27 

32 

8992 

goSo 

9107 

9i65 

9222 

9280 

27 

33 

8295 

8355 

84i4 

8474 

8533 

8593 

26 

33 

9337 

9395 

9452 

95ib 

9567 

9625 

26 

34 

8652 

8711 

8771 

883o 

889o 

8949 

25 

34 

9682 

974o 

9797 

9855 

9912 

9969 

25 

35 

9008 

9068 

9I27 

9186 

9246 

93o5 

24 

35 

9.640027 

oo84 

0142 

oi99 

O257 

o3i4 

24 

36 

9364 

9424 

9483 

9543 

96o2 

9661 

23 

36 

.  0371 

0429 

o486 

o544 

0601 

o658 

23 

37 

9720 

9780 

9839 

9898 

9958 

..17 

22 

37 

0716 

o773 

o83o 

0888 

0945 

1002 

22 

38 

9.620076 

oi36 

0195 

0254 

o3i3 

o373 

21 

38 

1060 

1174 

1232 

1289 

1  346 

21 

39 

0432 

0491 

o55o 

0610 

o669 

O728 

20 

39 

i4o4 

i46i 

i5i8 

i575 

i633 

1690 

2O 

4o 

9.620787 

o846 

0906 

o965 

1024 

io83 

I9 

4o 

9.641747 

i8o5 

1862 

1919 

19-76 

2034 

19 

4i 

Il42 

I2OI 

1261 

1320 

i379 

i438 

18 

4i 

2091 

2148 

22O5 

2263 

2320 

2377 

18 

42 

1497 

i556 

1616 

!675 

1734 

1-793 

17 

42 

2434 

249I 

2549 

2606 

2663 

2720 

1-7 

43 

1862 

1911 

i97o 

2029 

2088 

2l47 

16 

43 

2777 

2834 

2892 

2949 

36o6 

3o63 

16 

44 

2207 

2266 

2325 

2384 

2443 

2502 

i5 

44 

3l20 

3i77 

3235 

3292 

3349 

34o6 

i5 

45 

2.56i 

2620 

2679 

2738 

2797 

2856 

i4 

45 

3463 

3520 

3577 

3634 

369i 

3749 

i4 

46 

29!5 

2974 

3o33 

3092 

3i5i 

3210 

i3 

46 

38o6 

3863 

3920 

3977 

4o34 

4091 

i3 

47 

3260. 

3328 

3387 

3446 

35o5 

3564 

I  2 

47 

4i48 

42o5 

4262 

43i9 

4376 

4433 

12 

48 

3623 

3682 

374i 

38oo 

3858 

39i7 

ii 

48 

4490 

4547 

46o4 

466i 

47'i8 

4775 

II 

49 

3976 

4o35 

4o94 

4i53 

4212 

4271 

10 

49 

4832 

4889 

4946 

5oo3 

5o6o 

5n7 

IO 

5o 

9.62433o 

4388 

4447 

45o6 

4565 

4624 

9 

5o 

9.645174 

523i 

5288 

5345 

5402 

5459 

9 

5i 

4683 

4742 

48oo 

4859 

4918 

4977 

8 

5i 

55i6 

5573 

563o 

5687 

5744 

58oi 

8 

52 

5o36 

5094 

5i53 

5212 

5271 

533o 

7 

52 

5857 

59i4 

597i 

6028 

6o856i42 

7 

53 

5388 

5447 

55o6 

5565 

5623 

5682 

6 

53 

6199 

6256 

63i3 

6369 

6426  6483 

6 

54 

574i 

58oo 

5858 

5917 

5976 

6o35 

5 

54 

654o 

6597 

6654 

6710 

6-7676824 

5 

55 

6o93 

6i52 

6211 

6269 

6328 

6387 

4 

55 

6881 

6938 

6995 

7o5i 

-7108  7i65 

4 

56 

6445 

65o4 

6563 

6621 

6680 

6739 

3 

56 

7222 

7279 

7335 

7392 

7449  75c6 

3 

57 

6797 

6856 

69i5 

6973 

7o32 

7090 

2 

57 

7562 

76i9 

7676 

7733 

7789  7846 

2 

58 

7149 

7208 

7266 

7325 

7383 

7442 

I 

58 

79°3 

796o 

8016 

8o73 

8i3o|8i86 

I 

59 

75oi 

7559 

•7618 

7676 

7735 

7793 

O 

59 

8243 

83oo 

8356 

84i3 

84708526 

O 

60"     50" 

40" 

30" 

20" 

10" 

. 

60"     50" 

40" 

30"   20"   10" 

S" 

Co-tangent  of  67  Degrees. 

i 

, 

Co-tangent  of  66  Degrees. 

„  p,  J  1"  2"  3"  4"  5"  6"  7"  8"  9" 

.  (  1"  2"  3"  4"  5"  6"  7"  &'  9''' 

}  6  12  18  24  30  36  42  48  54 

trt£  6  12  17  23  29  35  40  46  52 

LOGARITHMIC    SINES. 


1 

Sine  of  24  Degrees. 

c  |      Sine  of  2;>  Degrees. 

Li 

0" 

10-' 

20" 

30" 

40" 

50" 

2, 

0" 

10"   20"   30" 

40'' 

50" 

o 

2 

9.609313 

9597 

9880 

9361 
9644 
9928 

94o8 
9691 
9975 

9455 
9739 

.  .22 

9502 
9786 
..69 

9550 

9833 
.116 

59 
58 
57 

o 

i 

2 

9.625948 

62I9 

649o 

5993(6039  6o84 

62^4  63o9  6354 
6535658o  6625 

6129 

64oo 
6670 

6174 
6445 
67i5 

59- 
58 
57 

3  9.610164 

0211 

0258 

o3oo 

o352 

0399 

56 

3 

6760 

68o5 

685o 

6895 

694o 

6985 

56 

4    o44y 

0494 

o54i 

o588 

o635 

0682 

55 

4 

7o3o 

7075 

7120 

7i65 

7210 

7255 

55 

V    0729 

0776 

0823 

0871 

0918 

0965 

54 

5 

73oo 

7345 

739° 

7435 

7480 

7525 

54 

3      1012 

1059 

1106 

n53 

1  200 

1247 

53 

6 

757o 

76i5 

7660 

77o5 

775° 

7795 

53 

m 

1294 

i34i 

i388 

i435 

1482 

1529 

52 

7 

784o 

7885 

7929 

7974 

8oi9 

8o64 

52 

1  8 

i576 

1623 

1670 

1717 

iy64 

1811 

5i 

8 

8io9 

8i54 

8i99 

8244 

8289 

8333 

5i 

9 

i858 

1905 

1952 

1999 

2046 

2093 

5o 

9 

8378 

8423 

8468 

85i3 

8558 

8602 

5o 

TO 

q.6;2i4o 

2187 

2234 

2280 

2327 

2374 

49 

10 

9.  628647 

8692 

8737 

8782 

8826 

8871 

49 

ii 

2421 

2468 

25i5!2562 

2609 

2655 

48 

ii 

89i6 

8961 

9006 

9o5o 

9o95 

9i4o 

48 

12 

2702 

2749 

2796  2843 

2890 

2936 

47 

12 

9i85 

9229 

9274 

93i9 

9363 

94o8 

47 

i3 

2983 

3o3o 

3077 

3i24 

3i7i 

3217 

46 

i3 

9453 

9498 

9542 

9587 

9632 

9676 

46 

t4 

3264 

33n 

3358 

34o4 

345i 

3498 

45 

i4 

9721 

9766 

9810 

9855 

99oo 

9944 

45 

i5 

3545 

SSgi 

3638  3685 

3732 

3778 

44 

i5 

9989 

..34 

..78 

.123 

.168 

.212 

44 

16 

3825 

3872 

39i83965 

4OI2 

4o58 

43 

16 

9.630257 

o3oi 

o346 

o39i 

0435 

o48o 

43 

*7 

4io5 

4i52 

41984245 

4292 

4338 

42 

J7 

o524 

o569 

o6i3 

o658 

0703 

o747 

42 

18 

4385 

4432 

44?8  4525 

457i 

46:8 

4i 

18 

0792 

o836 

0881 

O925 

o97o 

1014 

4i 

*9 

4665 

4711 

4758  48o4 

485i 

4898 

4o 

J9 

1069 

no3 

n48 

II92 

I237 

1281 

4o 

20 

9.614944 

4991 

5o37  5o84 

5i3o 

5177 

39 

20 

9.63i326 

1370 

i4i5 

i459 

i5o4 

1  548 

39 

21 

5223 

5270 

53:65363 

5409 

5456 

38 

21 

i593 

i637 

1681 

1726 

1770 

i8i5 

38 

22 

55o2 

5549 

55g5  5642 

5688 

5735 

37 

22 

i85g 

1904 

1948 

I992 

203712081 

37 

23 

5  78  1 

6828 

5874  5921 

5967 

6oi3 

36 

23 

2125 

2170 

22l4 

2259 

23o3 

2347 

36 

24 

6060 

6106 

6i53  6199 

6245 

6292 

35 

24 

2392 

2436 

2480 

2525 

2569 

2613 

35 

25 

6338 

6385 

643i  6477 

6524 

6570 

34 

25 

2658 

2702 

2746 

279° 

2835 

'2879 

34 

26 

6616 

6663 

6709,6755 

6802 

6848 

33 

26 

2923 

2968 

3012 

3o56 

3ioo 

3i45 

33 

27 

6894 

6941 

6987  7o33 

7080 

7126 

32 

27 

3189 

3233 

3277 

3322 

3366 

34io 

32 

28 

7172 

7218 

7265  7311 

735-7 

74o3 

3i 

28 

3454 

3498 

3543 

3587 

363i 

3675 

3i 

29 

745o 

7496 

7542  7588 

7635 

7681 

3o 

29 

37i9 

3764 

38o8 

3852 

3896 

394o 

3o 

309.617727 

7773 

7819  786617912 

7958 

29 

3o 

9.633984 

4028 

4o73 

4117 

4i6i 

42o5J29 

3i 

8oo4 

8o5o 

80968143 

8189 

8235 

28 

3i 

4249 

4293 

4337 

438i 

4426 

4470 

28 

32 

'  8281 

8327 

83788419 

8465 

85i2 

27 

32 

45i4 

4558 

4602 

4646 

469o 

4734 

27 

33 

8558 

86o4 

86508696 

8742 

8788 

26 

33 

4778 

4822 

4866 

4910 

4954 

4998 

26 

34 

8834 

8880 

8926  8972 

9018 

9o64 

25 

34 

5o42 

5o86 

5i3o 

5i74 

52i8 

5262 

25 

35 

9110 

9i56 

9202  9248 

9294 

934o 

24 

35 

53o6 

535o 

5394 

5438 

5482 

5526 

24 

36 

9386 

9432 

9478  9524 

9570 

96i6 

23 

36 

5570 

56i4 

5658 

5702 

5746 

579O 

23 

3? 

9662 

9708 

9754  9800 

9846 

9892 

22 

37 

•  5834 

5877 

5921 

5965 

6oo9 

6o53 

22 

38 

9938 

9984 

..3o!..76 

.121 

.167 

21 

38 

6097 

6i4i 

6:85 

6229 

6272 

63:6 

21 

39  9.620213 

0259 

o3o5o35i 

o397 

o443 

20 

39 

636o 

64o4 

6448 

649a 

6535 

6579 

2O 

409.620488 

o534 

o58o  0626 

0672 

0718 

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4o 

9.  636623 

6667 

67n 

6754 

6798 

6842 

«9 

4i 

o763 

oSoc, 

o855  0901 

0947 

0992 

18 

4i 

6886 

693o 

6973 

7017 

7061 

7io5 

18 

42 

io38 

1084  n3o!ii75 

1221 

1267 

J7 

4s 

7i48 

7192 

7236 

7280 

7323 

7367 

J7 

43    i3i3 

i358 

1  4o4  i45o|i496 

i54i 

16 

43 

74xi 

7455 

7498 

7542 

7586 

7629 

16 

44    i587 

i633 

1078  1724  1770 

1816 

i5 

44 

7673 

7717 

776o 

78o4 

7848 

7891 

i5 

45'    1861 

1907 

1953  1998 

2044 

2089 

i4 

45 

7935 

7979 

8022 

8066 

8110 

8i53 

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46    2i35 

2181 

2226,2272 

23i8 

2363 

i3 

46 

8197 

8240 

8284 

8328 

837i 

84:5 

i3 

4?    2409 

2454 

25oo  2546 

2591 

2637 

12 

47 

8458 

85o2 

8546 

8589 

8633 

8676  12 

48    «682 

2728 

27732819 

2865 

2910 

II 

48 

8720 

8763 

88o7 

885i 

8894 

8938  n 

4g    2966 

3ooi 

3o47 

3092 

3i38 

3i83 

10 

49 

8981 

9025 

9o68 

9112 

9i55 

9i99 

10 

5o  9.623229 

3274 

3320 

3365 

34u 

3456 

9 

5o 

9.639242 

9286 

9329 

9373 

94i6 

946o 

9 

5i 

35b2 

3547 

3593 

3638 

3683 

0729 

8 

5i 

95o3 

9546 

959o 

9633 

9677 

9720 

8 

52 

3774 

3820 

3865 

3911 

3956 

4ooi 

7 

52 

9764 

9807 

985i 

9894 

9937 

998i 

7 

53 

4o47 

4092 

4i38 

4i83 

4228 

4274 

6 

53 

9.640024 

0068 

OIII 

oi54 

oi98 

0241 

6 

54    43i9 

4364 

44io 

4455 

45oo 

4546 

5 

54 

0284 

o328 

o37i 

o4i4 

o458 

o5oi 

5 

55)    459i 

4636 

4682 

47274772 

48i8 

4 

55 

o544 

o588 

o63i 

0674 

o7i8 

0761 

4 

56    4863 

4908 

4954 

4999|5o44 

5o89 

3 

56 

0804 

o848 

o89i 

o934 

0978 

1021 

3 

5? 

5i35 

5i8o 

5225 

52706316 

536i 

2 

57 

1064 

1107 

n5i 

1194 

12  }7 

1280 

2 

58 

5406 

545  1 

5496 

5542|5587 

5632 

I 

58 

1324 

1367 

i4io 

i453 

i4g6 

i54o 

I 

59    5677 

5722 

5768 

58i35858 

59o3 

0 

59 

i583 

1626 

i669 

1712 

i756  1799 

0 

60"  .- 

50" 

40" 

30"  .  20" 

10" 

n 

60"    |  50"  |  40" 

30"   20"   10" 

d 

.  Co-  sine  of  65  Degrees. 

§ 

Co-sine  of  64  Degrees. 

§ 

(  1"  2"  3"  4"  5"  6"  7"  8"  9" 

p  p   (  1"  2"  3"  4"  5"  6"  7"  8"  9" 

r.Jartj  5   g  H  ig  23  2g  32  37  42 

1  I  4   9  13  18  22  26  31  35  40 

LOGARITHMIC    TANGENTS. 


1 

Tangent  of  24  Degrees. 

• 

Tangent  of  25  Degrees. 

9 

0"    |  10" 

20"  |  30" 

40" 

50" 

* 

0" 

10" 

20" 

30"  |  40"  1  50' 

o 

9.648583 

864o 

8696 

8753 

88lO 

8866 

59 

O 

9.  668673 

8728 

8782 

8837i88Q2  8947 

59 

2 

8923 
9263 

8980 
9319 

9o36 
9376 

9o93 
9433 

9i5o 
9489 

9206 
9546 

58 

2 

9002 
9332 

9°57 

9387 

9112 

9442 

9167(9222  9277 
9497(9552  9606 

58 

3 

9602 

g659 

97i5 

9772 

9829 

9885 

56 

3 

9661 

9716 

9771 

9826 

98819936 

56 

4 

9942 

9998 

..55 

.in 

.168 

.224 

55 

4 

9991 

..45 

.IOO 

.i55 

.210 

.265 

55 

5 

9.650281 

0337 

o394 

o45o 

0507 

o563 

54 

5 

9.670320 

0375 

0429 

o484 

o539 

o594 

54 

6 

0620 

0676 

o733 

o789 

o846 

O9O2 

53 

6 

0649 

0703 

o758 

o8i3 

0868 

0923 

53 

7 

0959 

ioi5 

IO72 

1128 

n85 

1241 

52 

7 

0977 

IO32 

1087 

1142 

1197 

I25l 

52 

8 

1297 

i354 

i4io 

i467 

i523 

i579 

5i 

8 

i3o6 

i36i 

1416 

1470 

i525 

i58o 

5i 

9 

1636 

1692 

i749 

i8o5 

1861 

1918 

5o 

9 

i635 

1689 

1744 

1799 

i853 

I9o8 

5o 

10 

9.651974 

2031 

2087 

2143 

2200 

2256 

49 

10 

9-67i963 

2018 

2072 

2127 

2182 

2236 

49 

ii 

23l2 

2369 

2425 

2481 

2538 

2594 

48 

ii 

229I 

2346 

2400 

2455 

25lO 

2564 

48 

12 

265o 

2707 

2763 

28l9 

2875 

2932 

47 

12 

26i9 

2674 

2728 

2783 

a838 

2892 

47 

i3 

2988 

3o44 

3ioi 

3i57 

32i3 

3269 

46 

i3 

2947 

3ooi 

3o56 

3m 

3i65 

322O 

46 

i4 

3326 

3382 

3438 

3494 

355i 

3607 

45 

i4 

3274 

3329 

3384 

3438 

3493 

3547 

45 

i5 

3663 

37i9 

3776 

3832 

3888 

3944 

44 

i5 

36o2 

3657 

3711 

3766 

3820 

3875 

44 

16 

4ooo 

4o57 

4n3 

4169 

4225 

4281 

43 

16 

3929 

3984 

4o38 

4o93 

4i48 

4202 

43 

»7 

4337 

4394 

445o 

45o6 

4562 

46i8 

42 

17 

43n 

4366 

4420 

4475 

4529 

42 

18 

4674 

473i 

4787 

4843 

4899 

4955 

4i 

18 

4584 

4638 

4693 

4747 

4802 

4856 

4i  • 

19 

Son 

5067 

5i23 

5179 

5236 

5292 

4o 

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49n 

4965 

5oi9 

5o74 

5i28 

5i83 

4o 

20 

9.655348 

54o4 

546o 

55i6 

5572 

5628 

39 

20 

9.675237 

5292 

5346 

54oi 

5455 

55o9 

39 

21 

5684 

5740 

5796 

5852 

5908 

5964 

38 

21 

5564 

56i8 

5673 

5727 

578i 

5836 

38 

22 

6020 

6076 

6  1  32 

6188 

6244 

63oo 

37 

22 

589o 

5945 

5999 

6o53 

6108 

6162 

37 

23 

6356 

6412 

6468 

6524 

658o 

6636 

36 

23 

6217 

6271 

6325 

638o 

6434 

6488 

36 

24 

6692 

6748 

68o4 

6860 

6916 

6972 

35 

24 

6543 

6597 

665i 

67o6 

6760 

68i4 

35 

25 

7028 

7084 

7140 

7196 

7252 

73o8 

34 

25 

6869 

6923 

6977 

7o32 

7086 

7140 

34 

26 

7364 

?4i9 

7475 

753i 

7587 

7643 

33 

26 

7194 

7249 

73o3 

7357 

7412 

7466 

33 

27 

7699 

7755 

7811 

7807 

7922 

7978 

32 

27 

7520 

7574 

7629 

7683 

7737 

7791 

32 

28 

8o34 

8090 

8i46 

8202 

8258 

83i3 

3i 

28 

7846 

7900 

7954 

8008 

8062 

8117 

3i 

29 

8369 

8425 

848  1 

8537 

8592 

8648 

3o 

29 

8i7i 

8225 

8279 

8334 

8388 

8442 

3o 

3o 

9.658704 

8760 

8816 

8871 

8927 

8983 

2O 

3o 

9.678496 

855o 

86o4 

8659 

8713 

8767 

29 

3i 

9039 

9095 

9i5o 

9206 

9262 

93i8 

2o 

3i 

8821 

8875 

8929 

8984 

9o38 

9092 

28 

32 

9373 

9429 

9485 

9540 

9596 

9652 

27 

32 

9i46 

9200 

9254 

93o8 

9363 

9417 

27 

33 

97o8 

9763 

9819 

9875 

993o 

9986 

26 

33 

9471 

9525 

9579 

9633 

9687 

974i 

26 

34 

9.660042 

0098 

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0209 

0265 

0320 

25 

34 

9795 

9849 

9904 

9958 

.  .  12 

..66 

25 

35 

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o43i 

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0598 

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24 

35 

9.680120 

0174 

0228 

0282 

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o39o 

24 

36 

0710 

o765 

0821 

0877 

0932 

0988 

23 

36 

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0660 

0714 

23 

37 

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1099 

n55 

1210 

1266 

1321 

22 

37 

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0822 

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0984 

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22 

38 

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1432 

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1  544 

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i655 

21 

38 

1092 

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1  200 

1254 

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21 

39 

1710 

1766 

1821 

1877 

1932 

1988 

20 

39     i4i6 

1470 

i524 

i578 

i632 

1686 

2O 

4o 

9.662043 

2099 

2i54 

22IO 

2265 

2321 

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4o 

9.68i74o 

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1  847 

1901 

1955 

2009 

J9 

4i 

2376 

2432 

2487 

2543 

2598 

2654 

18 

4i 

2o63 

2117 

2171 

2225 

2279 

2333 

18 

42 

2709 

2765 

2820 

2876 

2931 

2987 

17 

42 

2387J244o 

2494 

2548 

2602 

2656 

17 

43 

3o42 

3o97 

3i53 

3208 

3264 

33i9 

16 

43 

2710 

2764 

2817 

2871 

2925 

2979 

16 

44 

3375 

343o 

3485 

3541 

3596 

365i 

i5 

44 

3o33 

3o87 

3i4o 

3i94 

3248 

33o2 

i5 

45 

3707 

3762 

38i8 

3873 

3928 

3984 

i4 

45 

3356 

34io 

3463 

35i7 

357i 

3625 

i4 

46 

4o39 

4094 

4i5o 

42o5 

4260 

43i6 

i3 

46 

3679 

3732 

3786 

384o 

3894 

3947 

i3 

47 

437i 

4426 

4482 

4537 

4592 

4648 

12 

47 

4ooi 

4o55 

4109 

4162 

4216 

4270 

12 

48 

47o3 

4?58 

48i4 

4869 

4924 

4979 

II 

48 

4324 

4377 

443  1 

4485 

4539 

4592 

II 

49 

5o35 

5090 

5i45 

5200 

5256 

53n 

IO 

49 

4646 

4700 

4753 

4807 

486i 

4914 

10 

5o 

9,665366 

5421 

5477 

5532 

5587 

5642 

9 

5o 

9.684968 

5022 

5o75 

5129 

5i83 

5236 

9 

5i 

5698 

5753 

58o8 

5863 

59i8 

5974 

8 

5i 

5290 

5344 

5397 

545i 

55o5 

5558 

S 

52 

6o29 

6o84 

6i39 

6194 

6249 

63o5 

7 

52 

56i2 

5666 

57i9 

5773 

5827 

588o 

7 

53 

636o 

64i5 

6470 

6525 

658o 

6636 

6 

53 

5934 

5987 

6o4i 

6095 

6i48 

6202 

6 

54 

669i 

6746 

6801 

6856 

691  1 

6966 

5 

54 

6255 

63o9 

6363 

64i6 

6470 

6523 

5 

55 

7021 

7o76 

7132 

7187 

7242 

7297 

4 

55 

0577 

663o 

6684 

6737 

6791 

6845 

4 

56 

7352 

74o7 

7462 

7517 

7572 

7627 

3 

56 

6898 

6952 

7oo5 

7o59 

7112 

7166 

3 

57 

7682 

7737 

7792 

7847 

79°3 

7958 

2 

57 

7219 

7273 

7326 

738o 

7433 

7487 

2 

58 

8oi3 

8068 

8123 

8178 

8233 

8288 

I 

58 

7647 

7701 

7754 

7808 

I 

.1? 

8343 

8398 

8453 

85o8 

8563 

8618 

0 

59 

7861 

79i5 

7968 

8021 

8o75 

8128 

0 

60" 

50"   40" 

30"  1  20" 

10" 

. 

60" 

50"   40" 

30" 

20" 

10" 

g" 

Co-tangent  of  65  Degrees. 

§ 

Co-tangent  of  64  Degrees. 

p  p   $  1"  2"  3"  4"  5"  6"  7"  8"  9"  II  -p  -p   (  1"  2"  3"  4"  5"  6"  7"  8"  9" 

•  }  6  11  17  22  28  33  39  45  50  j  r-rartj  5   n  10  22  27  33  38  43  49 

50 


LOGARITHMIC    SINES. 


1 

Sine  of  26  Degrees. 

.3 

Sine  of  27  Degrees. 

S  f    0"    [  10"  |  20"  |  30"  i  40" 

50" 

& 

0" 

10' 

20''  |  30"~]  40"  1  50" 

O 

I 

9.641842 

2101 

|i885ii928  1971  2oi5 
214412187  223o  2273 

2o58 
2317 

59 

58 

o 
i 

9.657o47 
7295 

7088 
7336 

7129 

7377 

7171  7212 
74i8!746o 

?5o3 

59 

58 

2 

236o 

24o3  2446  24892532 

2575 

57 

2 

7542 

7584 

7625 

7666 

7707 

7749 

5? 

Q 

26i8'266i'27o4i2747;279o 

2833 

56 

3 

779° 

78^17872 

7913 

7955 

7996 

56 

4|    2877 

29202963 

3oo6  3o49 

3o92 

55 

4 

8037 

807818119 

8161 

8202 

8243 

55 

5    3i35 

3178  3221 

3264 

33o7 

335o 

54 

5 

8284 

8325 

836? 

84o8 

8449 

849o 

54 

6 

3393 

3436  3479 

3522 

3565 

36o7 

53 

6 

853i 

8572 

86i3 

8655 

8696 

8737 

53 

7 

365o 

3693|3736 

3779 

3822 

3865 

52 

7 

8778 

8819 

8860 

8901 

8942 

8983 

52 

8 

39o8 

4o37 

4123 

5i 

8 

9025 

9066 

9107 

9148 

9i89 

923o 

5i 

9 

10 

4i65 
9.  644423 

42o8'425i 
4465;45o8 

42  94  14  3  3  7l4  3  80 

455  1  14594)4637 

5o 
49 

9 

10 

9271 

9.6595i7 

93i293539374 
9558|9599!964o 

9435 
968i 

9476 
9722 

5o 

49 

ii 

468o 

4722476514808 

485i 

4894 

48 

ii 

9763 

98o4|9845 

9886 

9927 

9968 

48 

12 

4936 

4979  5022 

5o65 

5I08 

5i5o 

47 

12 

9  .660009 

oo5o 

0091 

Ol32 

0173 

02l4 

47 

i3 

5i93 

5236  5279 

532i 

5364 

54o7 

46 

i3 

0255 

0296 

o337 

o378|o4i9 

o46o 

46 

i4 

545o 

5492  5535 

5578 

5620 

5663 

45 

i4 

o5oi 

o54i 

o582 

0623 

o664 

0705 

45 

i5 

5706 

5749  5791 

5834 

58.77 

5919 

44 

i5 

0746 

0787 

0828 

0869 

o9o9 

0950 

44 

16 

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1570 

0 

60" 

50"  |  40"  |  30" 

20"  I  10" 

d 

60" 

50"   40" 

30" 

20" 

10" 

. 

Co-sine  of  63  Degrees. 

•9 

2 

Co-sine  of  62  Degrees. 

a 

P  Part  5  l"  ~"  3"  4"  5"  6"  7"  8/  9" 

p  p   (  1"  2"  3"  4"  5"  6"  7"  8"  9" 

I  4   8  13  17  21  25  30  34  38 

111  \  4   8  12  16  20  24  28  32  36 

L  O  G  A  R  I  T  11  M  I  C      T  A  N  G  E  N  T  S. 


1 

Tangent  of  26  Degrees. 

c 

Tangent  of  27  Degrees. 

s 

0" 

10"  !  20"  j  30"   40"   50" 

2 

0" 

10"  |  20" 

30" 

40" 

50-n 

0 

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2 

3 

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2 

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8 

52 

4663 

4715 

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4820 

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7 

52 

3232 

3283 

3334 

3385 

3436 

3487 

7 

53 

4976 

5029 

5o8i 

5i33 

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6 

53 

3538 

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369i 

3742 

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6 

54 
55 

5290 
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5342 
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5394 
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5498 
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555i 
5863 

5 
4 

54 
55 

3844 
4149 

3895 
4200 

3945 

3996 

4302 

40474098 
4353l44o3 

5 

4 

56 

59i6 

5968 

6020 

6072 

6124 

6176 

3 

56 

4454 

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4556 

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4658(4709 

3 

57 

6228 

6280 

6333 

6385 

6437 

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2 

57 

4760 

48io 

486i 

49I2 

4963  5oi4 

2 

58 

654i 

6593 

6645 

6697 

6749 

6801 

I 

58 

5o65 

5ii5 

5i66 

52I7 

5268  53i9 

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59 

6854 

6906 

6958 

7010 

7062 

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59 

5370 

5420 

547i 

5522 

55735624 

O 

60" 

50"  j  40" 

30" 

20" 

10" 

c 

60"    |  50" 

40"  |  30" 

20"   10" 

d 

Co-tangent  of  63  Degrees. 

i 

Co-tangent  of  02  Degrees. 

i 

p  Plrt<  I"  2"  3"  4"  5"  6"  7"  8"  9" 
1  r>  11  1G  21  2G  32  37  42  47 

,  (  I-1'  2"  3"  4"  5"  6"  7"  8"  9" 
trl{  5  10  15  21  20  31  36  41  46  j 

Li  0  G  A  R  I  T  H  M  I  C     SlNES. 


d 

Sine  of  28  Degrees. 

a 

Sine  of  29  Degrees. 

3 

ov 

10" 

20" 

30-7  |  40" 

50" 

3 

0"     IO5'   20" 

30"  '  40"  |  50" 

0 

9.671609 

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1688 

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1807 

59 

0 

9.68557i 

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59 

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1847 

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2005 

2045 

58 

I 

5799 

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595i 

5989 

58 

o 

2084 

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2242 

2282 

57 

2 

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3 

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2400 

2440 

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4 
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2558 
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54 

4 
5 

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6 

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52 

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8 

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8 

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9 

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18 

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5898 

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5976 

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18 

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6i33 

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36 

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36 

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3o 

9.692339 

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3i 

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3i 

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32 

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32 

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33 

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34 

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34 

323i 

3268 

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3342 

3379 

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25 

35 

9824 

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3453 

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4897 

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42 

1  443 

1482 

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42 

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5n8 

5i55 

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1828 

1866 

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43 

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5266 

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44 

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202O 

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44 

545o 

5487 

5524 

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2i35 

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2212 

225o 

2288 

2327 

i4 

45 

567i 

5708 

5745 

5782 

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5855 

i.4 

46 

2365 

24o3 

2442 

2480 

25l9 

2557 

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46 

5892 

5929 

5966 

6oo3 

6o39 

6076 

i3 

4? 

s595 

2633 

2672 

27IO 

2748 

2787 

12 

47 

6ii3 

6i5o 

6187 

6223 

6260 

6297 

12 

48 

2825 

2863 

29O2 

294o 

2978 

3oi6 

II 

48 

6334 

6370 

6407 

6444 

648  1 

65i7 

II 

49 

3o55 

3o93 

3i3i 

3i7o 

3208 

3246 

IO 

49 

6554 

659i 

6628 

6664 

67oi 

6738 

10 

5o 

9.683284 

3323 

336i 

3399 

3437 

3475 

9 

5o 

9.696775 

6811 

6848 

6885 

6921 

6958 

9 

5i 

35i4 

3552 

359o 

3628 

3667 

37o5 

8 

5i 

6995 

7o3i 

7068 

7io5 

7i4i 

7178 

8 

52 

3743 

378i 

38i9 

3858 

3896 

3934 

7 

52 

72i5 

725i 

7288 

7325 

736i 

7398 

7 

53 

3972 

4oio 

4o48 

4o87 

4i25 

4i63 

6 

53 

7435 

7471 

75o8 

7545 

758i 

76i8 

6 

54 

4201 

4239 

4277 

43i5 

4353 

4392 

5 

54 

7654 

769i 

7728 

7764 

7801 

7838 

5 

55 

443o 

4468 

45o6 

4544 

4582 

4620 

4 

55 

7874 

7911 

7947 

7984 

8020 

8o57 

4 

56 

4658 

4696 

4735 

4773 

48n 

4849 

3 

56 

8o94 

8i3o 

8167 

8203 

8240 

8276 

3 

57 

4887 

4925 

4963 

5ooi 

5o39 

5o77 

2 

57 

83-i  3 

8349 

8386 

8423 

8459 

8496 

7, 

58 

5n5 

5i53 

5i9i 

5229 

5267 

53o5 

I 

58 

8532 

8569 

86o5 

8642 

8678 

87i5 

: 

59 

5343 

538i 

54i9 

5457 

5495 

5533 

O 

59 

875i 

8788 

8824 

8861 

8897 

8934 

o 

60" 

50"  ; 

40" 

30"   20" 

10" 

c 

60" 

50" 

40" 

30"  |  20" 

19' 

d 

Co-sine  of  61  Degrees. 

s 

Co-sine  of  60  Degrees. 

1  ! 

p  p  .(  1"  2"  3"  4"  5"  6"  7"  8"  9" 

,  1"  2"  3"  4"  5"  6"  7"  8"  9"  1 

irt$  4  8  12  16  19  23  27  31  35 

irt\  4   7  11  15  19  22  2C  30  33  ) 

LOGARITHMIC    TANGENTS. 


53 


A 

Tangent  of  28  Degrees. 

c 

Tangent  of  29  Degrees. 

2 

Q1'    |  10" 

20"  |  30" 

40" 

50" 

« 

0"      10" 

20" 

30"  |  40" 

50" 

o 

I 

9.  725674  5725 
5979  6o3o 

5776  5827 

6081  6i3i 

5878 
6182 

5928 
6233 

D9 

58 

0 

I 

9.743752 
4o5o 

38o2 
4099 

385i 

4149 

SgOl 
4199 

395i 

4248 

4ooo 

4298 

II 

2 

3 

62846334 
6588  663Q 

63856436 
6690  6740 

6487 
6791 

6537 
6842 

56 

2 

3 

4348 
4645 

4397 
4695 

4447 
4744 

4496 
4794 

4546 
4844 

4596 
4893 

II 

4 
5 

6892 

7X97 

6943 

6994  7o45 
7298  7349 

7o95 
7399 

745o 

55 

54 

4 
5 

4943 
524o 

4992 
5290 

5339 

5092 

5389 

5i4i 
5439 

5i9i  55 
5488  54 

6 

755i 

76o2 

7653 

0 

77°3 

7754 

53 

6 

5538 

5587 

5637 

5686 

5736 

5785 

53 

7 

7805 

7855 

7906 

7957 

8007 

8o58 

52 

7 

5835 

5884 

5934 

5983 

6o33 

6082 

52 

8 

8109 

8i59 

8210 

8261 

83u 

8362 

5i 

8 

6i32 

6182 

623i 

6281 

633o 

638o 

5! 

9 

8412 

8463 

85i4J8564 

86i5 

8665 

5o 

9 

6429 

6479 

6528 

6577 

6627 

6676 

5o 

]  0 

9.728716 

8767 

8817 

8868 

8918 

8969 

49 

IO 

9.746726 

6775 

6825 

6874 

6924 

6973 

4o 

II 

12 

9020 
9  32  3 

9070 
9374 

9121 

9424 

9171 

94?5 

9222 
9525 

9272 
9576 

48 
47 

ii 

12 

7023 
73i9 

7072 
7369 

7122 

74i8 

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7468 

7221 
75i7 

7270 

7567 

48 
47 

1  3 

9626 

9677 

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9828 

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46 

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46 

1-4 

9929 

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45 

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8061 

8110 

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16 

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16 

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17 

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42 

17 

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9048 

42 

18 

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1  191 

1242 

1292 

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1393 

4i 

18 

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1  544 

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20 

9.731746 

1796 

1847 

1897 

1948 

1998 

39 

20 

9.749689 

9739 

9788 

9837 

9886 

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39 

21 

2048 

2099 

2149 

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2250 

2300 

38 

21 

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33 

22 

235! 

2401 

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2602 

37 

22 

9.  750281 

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23 

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23 

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36 

24 

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24 

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00-70 

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35 

25 

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3458 

35o8 

34 

25 

1167 

1216 

1265 

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1  364 

i4i3 

34 

26 

3558 

3609 

3659 

3709 

376o 

38io 

33 

26 

1462 

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1610 

i659 

1708 

33 

27 

386o 

3910 

396i 

4on 

4o6i 

4m 

32 

27 

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1806 

i856 

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1954 

2003 

32 

28 

4162 

4212 

4262 

43i2 

4363 

44i3 

3i 

28 

2O52 

2IOI 

2l5l 

22OO 

2249 

2298 

3i 

29 

4463 

45i3 

4564 

46i4 

4664 

47i4 

3o 

29 

2347 

2396 

2446 

2495 

2544 

2593 

3o 

3o 

9.734764 

48i5 

4865 

49  1  5 

4965 

5oi5 

29 

3o 

9.752642 

2691 

2740 

2789 

2839 

2888 

29 

3i 

5o66 

5n6 

5i66 

52i6 

5266 

53i7 

28 

3i 

2937 

2986 

3o35 

3o84 

3i33 

3i82 

28 

32 

5367 

54i7 

5467 

55i7 

5567 

56i8 

27 

32 

323i 

3280 

333o 

3379 

3428 

3477 

27 

33 

5668 

57i8 

5768 

58i8 

5868 

5918 

26 

33 

3526 

3575 

3624 

3673 

3722 

377i 

26 

34 

5969 

6019 

6069 

6119 

6169 

6219 

25 

34 

3820 

3869 

39i8 

3967 

4oi6 

4o66 

25 

35 

6269 

63i9 

6370 

6420 

6470 

652O 

24 

35 

4n5 

4i64 

42i3 

4262 

43n 

436o 

24 

36 

657o 

6620 

6670 

6720 

6770 

6820 

23 

36 

44o9 

4458 

45o7 

4556 

46o5 

4654 

23 

37 

6870 

6921 

6971 

7021 

7071 

7121 

22 

37 

4703 

4752 

48oi 

485o 

4899 

4948 

22 

38 

7171 

7221 

7271 

7321 

737i 

7421 

21 

38 

4997 

5o46 

5o95 

5i44 

5i93 

5242 

21 

39 

7471 

7521 

757i 

7621 

7671 

7721 

2O 

39 

5291 

534o 

5389 

5438 

5487 

5536 

20 

4o 

9.737771 

7821 

7871 

7921 

7971 

8021 

19 

4o 

9-755585 

5634 

5682 

573i 

5780 

582Q 

19 

4i 

8071 

8121 

8171 

8221 

8271 

832i 

18 

4i 

5878 

5927 

5976 

6o25 

6074 

6i23 

18 

42 

837i 

8421 

8471 

852i 

857i 

8621 

17 

42 

6172 

6221 

6270 

63i9 

6368 

64i6 

17 

43 

8671 

8721 

8771 

8821 

8871 

8921 

16 

43 

6465 

65i4 

6563 

6612 

6661 

6710 

16 

44 

8971 

9021 

9071 

9121 

9171 

9221 

i5 

44 

6759 

6808 

6857 

6905 

6954 

7oo3 

i5 

45 

9271 

9321 

9371 

9420 

9470 

9520 

i4 

45 

7052 

7IOI 

7i5o 

7199 

7247 

7296 

i4 

46 

47 

957o 
987o 

9620  9670 
9920  9969 

9720 
..19 

9770 
..69 

9820 
.  119 

i3 

12 

46 
47 

7345 
7638 

7394 
7687 

7443 
7736 

7492 
7785 

7834 

7589 
7882 

i3 

12 

48 

9.74oi69 

0219 

0269 

0319 

o368 

0418 

II 

48 

793i 

798° 

8029 

8078 

8127 

8i75 

II 

49 

o468 

o5i8 

o568 

0618 

0668 

0717 

IO 

49 

8224 

8273 

8322 

8371 

8419 

8468 

10 

5o 

9-740767 

0817 

0867 

0917 

0967 

1016 

9 

5o 

9.758517 

8566 

86i5 

8663 

8712 

8761 

9 

5i 

1066 

1116 

1166 

1216 

1265 

i3i5 

8 

5i 

8810 

8858 

8907 

8956 

9005 

9o53 

8 

52 

i365 

i4i5 

i465 

i5i4 

1  564 

1614 

7 

52 

9102 

9i5i 

9200 

9248 

9297 

9346 

7 

53 

1  664 

1714 

1763 

i8i3 

i863 

1913 

6 

53 

9395 

9443 

9492 

9541 

9590 

9638 

6 

54 

10,62 

2012 

2062 

2112 

2161 

221  I 

5 

54 

9687 

9736 

9785 

9833 

9882 

993i 

5 

55 

2261 

23ll 

236o 

24lO 

2460 

25lO 

4 

55 

9979 

..28 

••77 

.126 

•  174 

.223 

4 

56 

2559 

2609 

2659 

2709 

2758 

2808 

3 

56 

9.760272 

o32O 

o369 

o4i8 

o466 

o5i5 

3 

5- 

2858 

2907 

2957 

3007 

3o56 

3io6 

2 

57 

o564 

0612 

0661 

0710 

0758 

0807 

2 

58 

3i56 

3206 

3255 

33o5 

3355 

34o4 

I 

58 

o856 

0904 

0953 

1  002 

io5o 

1099 

I 

59 

3454 

35o4 

3553 

36o3 

3653 

3702 

O 

59 

n48 

1196 

1245 

1293 

1  342 

1391 

O 

60" 

50"  |  40" 

30" 

20" 

10" 

c 

60"    |  50" 

40" 

30" 

20" 

10" 

d 

Co-tangent  of  61  Degrees. 

Co-tangent  of  60  Degrees. 

1 

p  p   (  I"  2"  3"  4"  5"  6"  7"  8"  9" 
I  5  10  15  20  25  30  35  40  45 

p  p  .  <  1"  2"  3"  4"  5"  6//  7"  8//  9" 
irt\  5   10  15  20  25  29  34  39  44 

LOGARITHMIC    SINES. 


.3 

Sine  of  30  Degrees. 

a 

Sine  of  31  Degrees. 

"i 

% 

0" 

10" 

20"   30"   40" 

50" 

9 

0" 

10" 

20" 

30" 

40" 

50" 

0 

9.698970 

9006 

9043.  go79  9116 

9l52 

59 

o9.7n839 

i874 

I9o9 

i944 

IQ79 

20l4 

59 

I 

9189 

9225 

9262^298  9334 

937i 

58 

i 

2o5o 

2o85 

2120 

2i55 

2190 

2225 

58 

2 

9407 

9444 

9480  95i7  9553 

9589 

57 

2 

2260 

220)5 

2330 

2365 

24oo 

2434 

^7 

3 

9626 

9662 

9699  9735  977I 

98o8 

56 

0 

•j 

2469 

25o4 

2539 

2574 

26o9 

2644 

56 

4 

9844 

9880 

991-799539990 

..26 

55 

4 

2679 

2-714 

2749 

2784 

28i9 

2854 

55 

5 
6 

9.700062 
0280 

0099 
0317 

oi35  017110206 
o353o389o425 

0244 
0462 

54 
53 

5 
6 

2889 
3o98 

2924 

3i33 

2959 
3i68 

2994 

32o3 

3o29 
3238 

3o63 
3273 

54 
53 

7 

0498 

o534 

o57i  060-7 

oC43 

0680 

52 

7 

33o8 

3343 

3377 

3412 

344? 

3482 

52 

8 

0716 

0752 

o788 

0825 

0861 

o897 

5i 

8 

35i7 

3552 

3587 

362i 

3656 

369i 

5i  , 

9 

0933 

0970 

1006 

1042 

io78 

in5 

5o 

9 

3726 

376i 

3796 

383o 

3865 

39oo 

5o 

10 

9.  7on5i 

1187 

1223 

1259 

I296 

i332 

49 

IO 

9.7i3935 

397o 

4oo5 

4o39 

4o74 

4io9 

49 

ii 

1  368 

i4o4 

i44o 

1477 

i5i3 

i549 

48 

ii 

4i44 

4i79 

4213 

4248 

4283 

43i8 

48 

12 

i585 

1621 

i658 

1694 

i73o 

i766 

47 

12 

4352 

4387 

4422 

4457 

449i 

4526 

47 

i3 

1802 

i838 

i874 

1911 

i947 

i983 

46 

i3 

456i 

4596 

463o 

4665 

47oo 

4735 

46 

i4 

2019 

2o55 

2091 

2127 

2164 

22OO 

45 

i4 

4769 

48o4 

4839 

4873 

49o8 

4943 

45 

i5 

2236 

2272 

23o8 

2344 

238o 

2416 

44 

i5 

4978 

5oi2 

5o47 

5o82 

5n6 

5i5i 

44 

16 

2452 

2488 

2524|256i 

2597 

2633 

43 

16 

5i86 

522O 

5255 

529o 

5324 

5359 

43 

*7 

2669 

2705 

274i 

2777 

28i3 

2849 

42 

*7 

5394 

5428 

5463 

5498 

5532 

5567 

42 

18 

2885 

2921 

2957 

2993 

3o29 

3o65 

4i 

18 

56o2 

5636 

567i 

57o5 

574o 

5775 

4i 

*9 

3ioi 

3i37 

3i73 

32O9 

3245 

3281 

4o 

X9 

58o9 

5844 

5878 

59i3 

5948 

5982 

4o 

20 

9.7o33i7 

3353 

3389 

3425 

346i 

3497 

39 

20 

9.7i6oi7 

6o5i 

6086 

6121 

6i55 

6i9o 

39 

21 

3533 

3569 

36o5 

364i 

3677 

37i3 

38 

21 

6224 

6259 

6293 

6328 

6362 

6397 

38 

22 

3749 

3784  3820 

3856 

3892 

3928 

37 

22 

6432 

6466 

65oi 

6535 

657o 

66o4 

37 

23 

3964 

4ooo4o36 

4072 

4io8 

4i44 

36 

23 

6639 

6673 

67o8 

6742 

6777 

6811 

36 

24 

4179 

42i5 

425i 

4287 

4323 

4359 

35 

24 

6846 

6880 

69i5 

6949 

6984 

7oi8 

35 

25 

4395 

443  1 

4466 

45o2 

4538 

4574 

34 

25 

7o53 

7o87 

7I22 

7i56 

7i9i 

7225 

34 

26 

46io 

4646 

4682 

4717 

4753 

4789 

33 

26 

7259 

7294 

7328 

7363 

7397 

7432 

33 

27 

4825 

486i  4896 

4932 

4968 

5oo4 

32 

27 

7466 

75oo 

7535 

7569 

76o4 

7638 

32 

28 

5o4o 

5o75j5in 

5i47 

5i83 

52I9 

3i 

28 

7673 

77°7 

774i 

7776 

78io 

7844 

3i 

29 

5254 

529o  5326 

5362 

5397 

5433 

3o 

29 

7879 

79i3 

7948 

7982 

8016 

8o5i 

3o 

3o 

9.705469 

55o5  554o 

5576 

56i2 

5648 

29 

3o 

9.7i8o85 

8119 

8i54 

8188 

8223 

8257 

29 

3i 

5683 

57x96755 

579o 

5826 

5862 

28 

3i 

8291 

8326 

836o 

8394 

8429 

8463 

28 

32 

5898 

5933  5969 

6oo5 

6o4o 

6o76 

27 

32 

8497 

853i 

8566 

8600 

8634 

8669 

27 

33 

6112 

61476183 

62I9 

6254 

629o 

26 

33 

87o3 

8737 

8772 

8806 

884o 

8874 

26 

34 

6326 

636i  6397 

6433 

6468 

65o4 

25 

34 

8909 

8943 

8977 

9oi  i 

9o46 

9080 

25 

35 

6539 

6575  6611 

6646 

6682 

67i8 

24 

35 

9114 

9i48 

9i83 

9217 

925l 

9285 

24 

36 

6753 

678916824 

6860 

6895 

693i 

23 

36 

9320 

9354 

9388 

9422 

9456 

949i 

23 

3? 

6967 

7oo2  7o38 

7o73 

7io9 

7i45 

22 

37 

9525 

9559 

9593 

9627 

9662 

o696 

22 

38 

7180 

•7216  725i 

7287 

7322 

7358 

21 

38 

973o 

9764 

9798 

9833 

9867 

$9oi 

21 

39 

7393 

7429  7464 

75oo 

7535 

757i 

2O 

39 

9935 

9969 

...3 

..38 

..72 

.  106 

20 

4o 

9.707606 

•7642  7677 

77i3 

7748 

7784 

I  9 

4o 

9.72oi4o 

oi74 

0208 

0242 

0276 

o3n 

J9 

4i 

7810 

7855  789o 

7926 

796i 

7997 

18 

4i 

o345 

o379 

o4i3 

o447 

o48i 

o5i5 

18 

42;    8o32 

8068,8103 

8i39 

8i74 

8210 

!? 

42 

o549 

o583 

o6i7 

o652 

0686 

O72O 

J7 

43|    8245 

8j8o|83i6 

835i 

8387 

8422 

16 

43 

o754 

o788 

0822 

o856* 

o89o 

0924 

16 

44 

8458 

84938528 

8564 

8599 

8635 

i5 

44 

0958 

0992 

1026 

1060 

io94 

1128 

i5 

45 

8670 

87o5874i 

8776 

8811 

8847 

i4 

45 

1162 

1  196 

1230 

1264 

J298 

i332 

i4 

46 

8882 

89i8:8953 

8988 

9024 

9o59 

i3 

46 

1  366 

i4oo 

i434 

1  468 

i5o2 

i536 

i3 

4? 

9094 

9i3o  9i65 

92OO 

9236 

927I 

12 

47 

i57o 

i6o4 

i638 

l672 

1706 

i74o 

12 

48 

9306 

9342  9377 

94l2 

9448 

9483 

II 

48 

i774 

1808 

1842 

i876 

I9IO 

i944 

II 

49 

95i8 

95539589 

9624 

9659 

9695 

IO 

49 

i978 

2OI2 

2046 

2080 

2Il4 

2148 

IO 

5o 

9.709730 

9765  9800 

9836 

987i 

99o6 

9 

5o 

9.722l8l 

22l5 

2249 

2283 

2317 

235i 

9 

5i 

9941 

9977  •-12 

..47 

..82 

.118 

8 

5i 

2385 

24l9 

2453 

2487 

2520 

2554 

8 

52 

9.710153 

01880223 

O259 

0294 

o329 

7 

52 

2588 

2622 

2656 

2690 

2724 

2757 

7 

53 

o364 

0399  o435 

o47o 

o5o5 

o54o 

6 

53 

279I 

2825 

2859 

2893 

2927 

296o 

6 

54 

o575 

061  1  0646 

0681 

o7i6 

o75i 

5 

54 

2994 

3028 

3o62 

3o96 

3i3o 

3i63 

5 

55 

0786 

0822  o857 

o892 

0927 

0962 

4 

55 

3i97 

323: 

3265 

3299 

3332 

3366 

4 

56 

0997 

io32  io67 

no3 

n38  n73 

3 

56 

34oo 

3434 

3468 

35oi 

3535,3569 

3 

57 

1208 

1243  I278 

i3i3 

1  348  1  383 

2 

^7 

36o3 

3636|367o 

37o4 

3738  377i 

2 

58 
59 

1419 
1629 

i454  i4&9 
1  664  1699 

ID24 

i734 

i559  i594 

i769  i8o4 

I 
O 

58 
59 

38o5  3839  3873  39o6  394o  3974 
4oo7|4o4i  4o75  4io9  4i42  4i76 

I 

O 

60" 

50"   40"   30"  |  20"   10" 

a 

60"     50"  |  40"   30"  |  20"   10"  |  Q. 

Co-sine  of  59  Degrees. 

'£ 

Co-sine  of  58  Degrees.     & 

f<  1"  '2"  3"  4"  5"  6"  7"  8"  9"  jl      .  <  1"  2"  3"  4"  5"  6"  7"  8"  9" 
1  I  4   7  11  11  18  21  25  29  32   A-lmt}  3   7   10  14  17  21  24  27  31 

LOGARITHMIC     TANGENTS. 


d  |    Tangent  of  30  Degrees. 

g 

Tangent  of  3  1  Degrees. 

*  1    0" 

10" 

20"   30" 

40* 

50" 

i 

0"    |  10" 

207 

30" 

40"  \  50" 

o  9.76i439 

1  488 

i537 

i585 

i634 

1682 

59 

0 

9.778774 

8821 

8869 

8917 

8964)9012 

59 

i 

I73l 

1-780 

1828 

1877 

1925 

1974 

58 

I 

9o6o 

9108 

9i55 

9203 

925: 

9298 

5s 

2 

2023 

2071 

2I2O 

2168 

22I7 

2266 

57 

2 

9346 

9394 

q44i 

Q489 

9537 

9584 

5? 

3j    23i4 

2363 

2411 

2460 

25o8 

2557 

56 

3 

9632 

9679 

9727 

9775 

9822 

9870 

56 

4   2606 

2654 

27o3 

2751 

28002848 

55 

4 

99i8 

9965 

.  .  i3 

..61 

.108 

.i56 

55 

5j    2897 

2945 

2994 

3o43 

3o9i 

3i4o 

54 

5 

9.780203 

O25l 

0299 

o346 

o394 

o44i 

54 

6    3i88 

3237 

3285 

.3334 

3382 

343  1 

53 

6 

0489 

o537 

o584 

o632 

0679 

0727 

53 

7 

3479 

352« 

3576 

3625 

3673 

3722 

52 

7 

0775 

0822 

0870 

0917 

o965 

ioi3 

52 

8 

377o 

38i9 

3867 

39i6 

3964 

4oi3 

5i 

8 

1060 

1108 

n55 

1203 

1250 

1298 

5i 

9 

4o6i 

4no 

4i58 

4207 

4255 

43o4 

5o 

9 

1  346 

i393 

i44i 

i488 

i536 

i583 

5o 

10 

9.764352 

44oo 

4449 

4497 

4546 

4594 

49 

10 

9.781631 

1678 

1726 

1774 

1821 

1869 

49 

r  r 

4643 

469i 

474o 

4788 

4836 

4885 

48 

ii 

1916 

1964 

2OII 

2o5g 

2106 

2i54 

48 

[2 

4933 

4982 

5o3o 

0079 

5l27 

5i75 

47 

12 

2201 

2249 

2296 

2344 

2391 

2439 

47 

r3 

5224 

5272 

532i 

5369 

54i8 

5466 

46 

i3 

2486 

2534 

258i 

2629 

2676 

2724 

46 

i4 

55i4 

5563 

56n 

566o 

57o8 

5756 

45 

i4 

2771 

2819 

2866 

2914 

2961 

3009 

45 

i5 

58o5 

5853 

59oi 

595o 

5998 

6047 

44 

i5 

3o56 

3io4 

3i5i 

3l99 

3246 

3294 

44 

16 

6o95 

6i43 

6l92 

6240 

6288 

6337 

43 

16 

334i 

3388 

3436 

3483 

353i 

3578 

43 

17 

6385 

6433 

6482 

653o 

6578 

6627 

42 

17 

3626 

3673 

3721 

3768 

38i6 

3863 

42 

18 

6675 

6723 

6772 

6820 

6868 

6917 

4i 

18 

3910 

3958 

4oo5 

4o53 

4ioo 

4r48 

4i 

19 

6965 

70i3 

7062 

7110 

7i58 

7207 

4o 

19 

4i95 

4242 

4290 

4337 

4385 

4432 

4o 

20 

9.767255 

73o3 

7352 

74oo 

7448 

7496 

39 

20 

9.784479 

4527 

4574 

4622 

4669 

4716 

39 

21 

7545 

7593 

7641 

769o 

7738 

7786 

38 

21 

4764 

48xi 

4859 

4906 

4953 

5ooi 

38 

22 

7834 

7883 

793i 

7979 

8027 

8076 

37 

22 

5o48' 

5o95 

5i43 

Sigo 

5238 

5285 

37 

23 

8124 

8172 

8221 

8269 

83i7. 

8365 

36 

23 

5332 

538o 

5427 

5474 

5522 

5569 

36 

24 

84i4 

8462 

85io 

8558 

8606 

8655 

35 

24 

56i6 

5664 

57n 

5758 

58o6 

5853 

35 

25 

8703 

875i 

8799 

8848 

8896 

8944 

34 

25 

Sgoo 

5948 

5995 

6042 

6090 

6137 

34 

26 

8992 

9o4o 

9o89 

9i37 

9i85 

9233 

33 

26 

6i84 

6232 

6279 

6326 

6374 

6421 

33 

27 

9281 

933o 

9378 

9426 

9474 

9522 

32 

27 

6468 

65i6 

6563 

6610 

6657 

67o5 

32 

28 

957i 

96i9 

9667 

97i5 

9763 

9811 

3i 

28 

6752 

6799 

6847 

6894 

694i 

6988 

3  1 

29 

9860 

99o8 

9956 

...4 

..52 

.  IOO 

3o 

29 

7o36 

7083 

7i3o 

7178 

7225 

7272 

3o 

3o 

9.770148 

oi97 

O245 

0293 

o34i 

o389 

29 

3o 

9-7873I9 

7367 

74i4 

746  1 

75o8 

7556 

20 

3i 

o437 

o485 

o534 

o582 

o63o 

0678 

28 

3i 

76o3 

765o 

7697 

7745 

7792 

7839 

28 

32 

0726 

0774 

0822 

o87o 

o9i9 

0967 

27 

32 

7886 

7934 

7981 

8028 

8o75 

8I22J27 

33 

ioi5 

io63 

mi 

nSg 

1207 

1255 

26 

33 

8170 

8217 

8264 

83n 

8359 

84o6 

20 

34 

i3o3 

i35i 

l399 

i448 

1496 

1  544 

25 

34 

8453 

85oo 

8547 

8595 

8642 

8689 

25 

35 

l592 

i64o 

1688 

i736 

1784 

i832 

24 

35 

8736 

8783 

883o 

8878 

8925 

8972 

24 

36 

1880 

I928 

1976 

2024 

2072 

2I2O 

23 

36 

9019 

9066 

9114 

9161 

9208 

9255 

23 

3? 

2168 

2216)2264 

23l2 

236i 

2409 

22 

37 

9302 

9349 

9397 

9444 

9491 

9538 

22 

38 

2457 

25o5l2553 

26OI 

2649 

2697 

21 

38 

9585 

9532 

9679 

9727 

9774 

982J 

21 

39 

2745 

2793 

2841 

2889 

2937 

2985 

2O 

39 

9868 

99  1  5 

9962 

...9 

..57 

.104 

2O 

4o 

9-773o33 

3o8i 

3129 

3i77 

3225 

3273 

X9 

4o 

9.790151 

0198 

O245 

0292 

0339 

o386 

I9 

4i 

332i 

3369 

34i7 

3465 

35i2 

356o 

18 

4i 

o434 

o48i 

o528 

o575 

0622 

o669 

18 

42 

36o8 

3656 

3704 

3752 

38oo 

3848 

17 

42 

0716 

o763 

0810 

0857 

0905 

O952 

17 

43 

3896 

3944 

3992 

4o4o 

4o88 

4i36 

16 

43 

0999 

io46 

1093 

n4o 

1187 

1234 

16 

44 

4i84 

4232 

4280 

4328 

4375 

4423 

i5 

44 

1281 

i328 

i375 

1422 

1469 

i5i6 

i5 

45 

4471 

45i9 

4567 

46i5 

4663 

4711 

i4 

45 

i563 

1611 

i658 

1705 

1752 

i799 

i4 

46 

4759 

4807 

4855 

4902 

495o 

4998 

i3 

46 

-846 

i893 

1940 

1987 

2034 

2081 

i3 

47 

5o46 

5og4 

5i42 

Sigo 

5238 

5286 

12 

47 

u*a 

2175 

2222 

2269 

23i6 

2363 

12 

48 

5333 

538i 

5429 

5477 

5525  5573 

II 

48 

2410 

2457 

25o4 

255i 

2598 

2645 

II 

49 

"  562i 

5668 

57i6 

5764 

58i2 

586o 

10 

49 

2692 

2739 

2786 

2833 

2880 

2927 

IO 

5o 

9.775908 

5956 

6oo3 

6o5i 

6099 

6i47 

9 

5o 

9.792974 

3021 

3o68 

3n5 

3i62 

3209 

9 

5i 

6i95 

6243 

6290 

6338 

6386 

6434 

8 

5i 

3256 

33o3 

335o 

3397 

3444 

3491 

8 

52 

6482  6529 

6577 

6625 

6673 

6721 

7 

52 

3538 

3585 

3632 

3679 

3726 

3773 

7 

53 

6768 

6816 

6864 

6912 

6960 

7007 

6 

53 

38i9 

3866 

39i3 

4oo7l4o54 

6 

54 

7o55 

7io3 

7161 

7r99 

7246 

7294 

5 

54 

4ioi 

4i48 

4i95 

4242 

4289'4336 

5 

55 
56 

57 
58 
59 

7628 
79i5 
8201 

8488 

739o 
7676 
7963 

8249 
8535 

7437 
7724 
8010 
8297 
8583 

7485 

7772 
8o58 

8344 
863i 

7533 
7819 
8106 
8392 

8678 

758i 
7867 
8  1  54 
844o 
8726 

4 
3 

2 

I 
O 

55 
56 

57 
58 

59 

4383 
4664 
4946 
5227 
55o8 

443o  4476 
47II4758 
4992  SoSg 
5274532i 
5555J5602 

452345704617 
48o5  4852  4899 
5o865i33:5i8o 
5367  54i4546i 
56495696  5742 

4 
3 

2 
I 
O 

00"     50"   40" 

30"  |  20" 

10" 

q 

60"     50"  |  40"  l  30"  |  20"  |  10" 

a 

Co-tangent  of  59  Degrees. 

.£* 

Co-tangent  of  58  Degrees. 

i 

P  Tart4  '"  ~"  :i"  4"  5"  °"  7"  8"  9" 

(  I"  2"  3"  4"  5"  6"  7"  8"  9" 

J  5  10  14  1!)  24  20  34  3f)  43 

in{  5   0   14  19  ^t  28  33  38  43 

56 


LOGARITHMIC    SINES. 


p  1      Sine  of  32  Degrees. 

d 

Sine  of  33  Degrees. 

m 

0 

10" 

20" 

30" 

40" 

50" 

ii 

0" 

10"   20" 

30" 

40" 

50" 

o 

9.724210 

4243 

4277 

43ri 

4344 

4378 

59 

o 

9>736io9 

6i4i 

6174 

6206 

6288 

62-71 

59 

il   44i2 

4445 

4479 

45:3 

454G 

458o 

58 

I 

63o3 

6336 

6368 

64oo 

6433 

6465 

58 

c    46i4 

464? 

468i 

47i5 

4748 

4782 

57 

2 

0498 

653o 

6562 

6595 

6627 

6659 

57 

3j    48  1  6 

4849 

4883 

4917 

495o 

4984 

56 

3 

6692 

6724 

6757 

6789 

6821 

6854 

56 

4j    5oi7 

5o5i 

5o85 

5n8 

5  1  52 

5i85 

55 

4 

6886 

6918 

695i 

6983 

7oi5 

7o48 

55 

5    6219 

5253 

5286 

5320 

5353 

5387 

54 

5 

7080 

7112 

7i45 

7i77 

7209 

724l 

54 

6 

5420 

5454 

5488 

552i 

5555 

5588 

53 

6 

7274 

73o6 

7338 

737i 

74o3 

7435 

53 

7 

5622 

5655 

5689 

572-3 

5756 

5789 

52 

7 

7467 

75oo 

7532 

7564 

7597 

7629 

5a 

8 

5823 

5856 

5890 

5923 

5957 

599o 

5i 

8 

7661 

7693 

7726 

7758 

779° 

•7822 

5i 

9 

6024 

6o57 

6091 

6124 

6i58 

6i9i 

5o 

9 

7855 

7887 

7919 

795i 

7983 

8016 

5o 

109.726226 

6258 

6292  6325 

6359 

6392 

49 

10 

9.73-8048 

8080 

8112 

8i45 

8i77 

8209 

49 

ii!    6426 

6459 

6493  6526 

656o 

6593 

48 

ii 

8241 

8273 

83o6 

8338 

837o 

8402 

48 

12      6626 

6660 

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6 

53 

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6 

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54 

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55 

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2 

57 

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69    5914 

5947 

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59 

7374 

7406 

7437 

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0 

|    60-' 

50" 

40" 

30"  |  20" 

10" 

c" 

60'' 

50" 

40" 

30"   20"  |  10" 

c 

Co-sine  of  57 

Degrees. 

^ 

Co-sine  of  56  Degices. 

§ 

C  1"  2"  3"  4" 

5"  6"  7"  8"  9" 

.  C  1"  2"  3"  4"  5"  6"  7''  8"  9" 

P.  Part  j  3  ?  1Q  13 

17  20  23  26  30 

irt\  3   6  10  13  16  19  22  25  29 

Lt  O  G  A  R  I  T  H  M  1  i:      TANGENTS. 


57 


'   C 

|    Tar 

0" 

igent  of  32  Degrees. 

a 
r£ 

Tangent  of  33  Degrees. 

10" 

20"  |  30" 

40" 

50" 

0" 

10"   20" 

30" 

40" 

SO" 

09.796789 

5836 

5883 

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49 

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6 

54 

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54 

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7397 

7442 

7488 

7533 

7579 

5 

55 

1  1  34 

1180 

1226 

1272 

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1  364 

4 

55 

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4 

56 

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1  549 

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3 

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7942 

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57 

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2 

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1 

59 

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2287 

2333 

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2426 

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8760 

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8861 

8897 

8942 

0 

60" 

50" 

40" 

30" 

20" 

10" 

71 

60" 

50" 

40" 

oO" 

20" 

10" 

j 

Co-tangent  of  57  Degrees. 

2 

Co-tangent  of  56  Degrees. 

2 

P  Part  5  ]//  ~"  3//  4"  5//  6"  7//  8//  9"  il      tf  l"  2"  3//  4//  5"  6"  7//  8>  9" 
l{  5   9  14  19  23  28  33  37  42  ||  1-rart{  5   9  14  J8  23  27  32  ,17  41 

58 


LOGARITHMIC    SINES. 


c      Sine  of  34  Degrees. 

.3 

Sine  of  35  Degrees. 

r~ 

r*. 

0" 

10" 

20"  |  30" 

40" 

50" 

m 

0" 

10" 

20"  [  30''  ]  40" 

50" 

O 

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7655 

7686 

7718 

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7874 

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58 

i 

8772 

8802 

8832 

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8922 

58 

0 

7936 

7967 

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2 

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3 

4 

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56 
55 

3 
4 

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5 

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5 

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6 

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8 

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5i 

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9522 

9553 

9584 

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IO 

9.760390 

0420 

o45o 

o48o 

o5o9 

o539 

49 

ii 

96i5 

96469677 

9708 

9739 

977° 

48 

ii 

o569 

o599 

0629 

o659 

o689 

o7i8 

48 

12 

98oi 

9832 

9863 

9894 

9925 

9956 

47 

12 

o748 

0778 

0808 

o838 

0868 

0898 

4? 

i3 

9987 

..18 

..48 

••79 

.110 

.i4i 

46 

i3 

0927 

°957 

0987 

IOI7 

io47 

io76 

46 

i4 

9.750172 

0203 

0234 

0265 

O296 

0327 

45 

i4 

1106 

n36 

1166 

1196 

1225 

1255 

45 

i5 

o358 

o389 

0420 

o45  1 

0482 

o5i2 

44 

i5 

1285 

i3i5 

1  345 

i374 

i4o4 

i434 

44 

1.6 

o543 

o574 

o6o5 

o636 

o667 

o698 

43 

16 

1  464 

i494 

i523 

i553 

!583 

i6i3 

43 

ll 

o729 

0760  070,1 

0821 

o852 

o883 

42 

«7 

1642 

1672 

1702 

I732 

1761 

1791 

42 

18 

o9i4 

o945  o976 

1007 

io37 

1068 

4i 

18 

1821 

i85i 

1880 

1910 

i94o 

1969 

4i 

J9 

io99 

u3o 

1161 

1192 

1222 

1253 

4o 

J9 

i999 

2O29 

2059 

2088 

2118 

2148 

4o 

20 

9.75i284 

i*3z5 

1  346 

i377 

i4o7 

i438 

39 

20 

9.  762177 

2207 

2237 

2267 

2296 

2326 

39 

21 

i469 

i5oo 

i53i 

i56i 

l592 

1623 

38 

21 

2356 

2385 

24i5 

2445 

2474 

25o4 

38 

22 

i654 

i685 

I7i5 

i746 

I777 

1808 

37 

22 

2534 

2563 

2593 

2623 

2652 

2682 

37 

23 

i839 

i869 

I900 

1931 

I962 

I992 

36 

23 

2712 

2741 

2771 

2801 

283o 

2860 

36 

24 

2023 

2o54 

2o85 

2Il5 

2146 

2I77 

35 

24 

2889. 

29I9 

2949 

2978 

3oo8 

3o38 

35 

25 

22O8 

2238 

2269 

23oo 

233o 

236i 

34 

25 

3067 

3o97 

3i26 

3i56 

3i86 

32i5 

34 

26 

2392 

2423 

2453 

2484 

25i5 

2545 

33 

26 

3245 

3274 

33o4 

3333 

3363 

3393 

33 

27 

2576 

2607 

26372668 

2699 

2729 

32 

27 

3422 

3452 

348i 

35n 

354o 

357o 

3^ 

28 

2760 

279I 

2822 

2852 

2883 

29l4 

3i 

28 

36oo 

3629 

3659 

3688 

37i8 

3747 

3i 

29 

2944 

2975 

3oo5 

3o36 

3o67 

3o97 

3o 

29 

3777 

38o6 

3836 

3865 

3895 

3925 

3o 

3o 

9.753128 

3i59 

3i89 

3220 

325i 

3281 

29 

3o 

9.763954 

3984 

4oi3 

4o43 

4072 

4102  29 

3i 

33i2 

3342 

3373 

34o4 

3434 

3465 

28 

3i 

4i3i 

4i6i 

4190 

4220 

4249 

4279 

28 

32 

3495 

3526 

3557 

3587 

36i8 

3648 

27 

32 

43o8 

4338 

4367 

4396 

4426 

4455 

27 

33 

3679 

3710 

374o 

377i 

38OI 

3832 

26 

33 

4485 

45i4 

4544 

4573 

46o3 

4632 

26 

34 

3862 

3893 

3923 

3954 

3985 

4oi5 

25 

34 

4662 

469i 

4720 

475o 

4779 

4809 

25 

35 

4o46 

4076  4107 

4i37 

4i68 

4i98 

24 

35 

4838 

4868 

4897 

4926 

4956 

4985 

24 

36 

4229 

4269  4290 

4320 

435i 

438i 

23 

36 

5oi5 

5o44 

5o74 

5io3 

Dl32 

5i62 

23 

3? 

44i2 

4442 

4473 

45o3 

4534 

4564 

22 

37 

Sigi 

5221 

525o 

5279 

53o9 

5338 

22 

38 

4595 

4625 

4656 

4686 

47i7 

4747 

21 

38 

5367 

5397 

5426 

5456 

5485 

55i4 

21 

39 

4778 

48o8 

4839 

4869 

49oo 

493o 

2O 

39 

5544 

5573 

56o2 

5632 

566i 

56go 

20 

4o 

9.7549,60 

4991 

5O2I 

5o52 

5o82 

5n3 

I  9 

4o 

9.765720 

5749 

5778 

58o8 

5837 

5866 

1  9 

4i 

5i43 

5i73 

52o4 

5234 

5265 

5295 

18 

4: 

5896 

5925 

5954 

5984 

6oi3 

6042 

18 

42 

5326 

53565386 

54i7 

5447 

5478 

*7 

42 

6072 

6101 

6i3o 

6i59 

6189 

6218 

17 

43 

55o8 

5538 

5569 

5599 

5629 

566o 

16 

43 

6247 

6277 

63o6 

6335 

6364 

6394 

16 

44 

56go 

5721 

575i 

578i 

58125842 

i5 

44 

6423 

6452 

648  1 

65n 

654o 

6569 

i5 

45 

5872 

SgoS 

5933 

5963 

5994  6024 

i4 

45 

6598 

6628 

6657 

6686 

6715 

6745 

i4 

46 

6o54 

6o85 

6n5 

6i45 

6176  6206 

i3 

46 

6774 

68o3 

6832 

6862 

6891 

6920 

i3 

47 

6236 

6267 

6297 

6327 

6358  6388 

12 

47 

6949 

6978 

7oo8 

7o37 

7066 

7o95 

12 

48 

64i8 

6448 

6479 

65o9 

6539657o 

II 

48 

7124 

7i54 

7i83 

72I2 

7241 

7270 

I  I 

49 

6600 

663o  6660 

669i 

672i  675i 

IO 

49 

73oo 

7329 

7358 

7387 

74i6 

7445 

10 

5o 

9.756782 

6812 

6842 

6872 

69o3  6933 

9 

5o 

9.767475 

75o4 

7533 

7562 

759i 

7620 

9 

5i 

6963 

6993 

7023 

7o54 

7o84  7n4 

8 

5i 

7649 

7679 

77o8 

7737 

7766 

7795 

8 

5i    7i44 

7175 

72o5 

7235 

7265  7295 

7 

52 

7824 

7853 

7882 

79I2 

794i  797o 

7 

53*    7326 

7356 

7386 

74i6 

7446  7477 

6 

53 

7999 

8028 

8o57 

8086 

8n58i44 

6 

54 

75°7 

75377567 

7597 

76277658 

5 

54 

8173 

82o3 

8232 

826l 

82908319 

5 

55 

7688 

7718, 

7748 

7778 

78087839 

4 

55 

8348 

8377 

84o6 

8435 

8464  84g3 

4 

56 

7869 

7899 

7929 

7959 

7989  8019 

3 

56 

8522 

855i 

858o 

8600 

86388668 

3 

57 
58 

8o5o 
823o 

8o8o8iio|8i4o 
8260829018321 

8170  8200 
835i'838i 

2 
I 

57 
58 

8697 

8871 

87268755 
89oo  8929 

8784 
8958 

88i38842 
898-7  9016 

2 

I 

59 

3411 

844i 

8471  85oi 

853i  856i 

O 

59 

9o45j9o74  9io3  9i3s 

9161  9i9o 

0 

60"    |  50"  | 

40"  |  30"  j  20"   10" 

ff 

60"    1  50'   40"  |  30"  |  20"  |  10" 

C 

Co-sine  of  55  Degrees. 

Jo 

% 

Co-sine  of  54  Degrees.     ~ 

C  t"  2"  3"  4"  5"  6"  7"  8"  9" 
irl£  3   6   9  12  15  18  21  25  28 

P  P-,rt5  l"  ~"  3"  4//  5"  fi"  ?'   8''  9" 

.\:Z   0   9   12  15  18  21  24  27 

LOGARITHMIC    TANGENTS. 


50 


jj 

Tangent  of  34  Degrees. 

a  \    Tangent  of  35  Degrees. 

* 

0"    |  10"  1  20"  |  30" 

40"  |  50" 

3  j    0"    |  10"   20" 

30"  |  40"  |  50" 

0 

I 

57128987 

9,260 

9033(9078 

93o5935i 

9I24 

9396 

9i69  92i5 
944^  9487 

59 

58 

O 
I 

9.845237 

5496 

554o 

53i6 

5585 

536i 
563o 

5406 

5675 

545  1 

5720 

59 

58 

2 

9532 

9578 

9623 

9669 

97i4 

9759 

57 

2 

5764 

58o9 

5854 

5899 

5944 

5988 

57 

3 

9805 

985o 

9895  <;94i 

9986 

..32 

56 

3 

6o33 

6078 

6i23 

6168 

6212  6257 

56 

4 
5 

9.830077 
0349 

OI22 

o395 

oi68|o2i3 
o44o  o485 

0258 
o53i 

o3o4 
o576 

55 

54 

4 
5 

63o2 
657o 

6347 
66i5 

6660 

6436 
67o5 

648i 
675o 

6526 

ij 

6 

0621 

0667 

0712 

o757 

o8o3 

o848 

53 

6 

6839 

6884 

6929 

6973 

7oi8 

7o63 

53 

7 

0893 

o939 

o984 

I029 

io75 

II2O 

52 

7 

7108 

7l52 

7J97 

7242 

7287 

733i 

5a 

8 

n65 

I2II 

1256 

i3oi 

1  347 

i392 

5i 

8 

7370 

7421 

7465 

75io 

7555 

76oo 

5i 

9 

i437 

i483 

i528 

i573 

i6i9 

1  664 

5o 

9 

7644 

7689 

7734 

7779 

7823 

7868 

5o 

10 

9.831709 

i755 

1800 

i845 

1891 

i936 

49 

10 

9.847913 

7957 

8002 

8o47 

8o92 

8i36 

49 

ii 

1981 

2026 

2072 

2II7 

2162 

2208 

48 

ii 

8181 

8226 

8270 

83i5 

836o  84o5 

48 

12 

2253 

2298 

2343 

2389 

2434 

2479 

47 

12 

8449 

8494 

8539 

8583 

8628  8673 

47 

i3 

2525 

2570 

26i5 

2660 

27o6 

275l 

46 

i3 

8717 

8762 

88o7 

885i 

8896 

894i 

46 

i4 

2796 

2842 

2887 

2932 

2977 

3o23 

45 

i4 

8986 

9o3o 

9o75 

9I20 

9i64 

9209 

45 

i5 

3o68 

3n3 

3i58 

3204 

3249 

3294 

44 

i5 

9254 

9298 

9343 

9388 

9432 

9477 

44 

16 

3339 

3385 

343o 

3475 

3520 

3566 

43 

16 

9522 

9566 

96n 

9656 

97oo 

9745 

43 

17 

36n 

3656 

3701 

3747 

3792 

3837 

42 

17 

979° 

9834 

9879 

9924 

9968 

..i3 

42 

18 

3882 

3927 

3973 

4oi8 

4o63 

4io8 

4i 

18 

9.85oo57 

OIO2 

0147 

oi9i 

0236 

0281 

4i 

I9 

4i54 

4199 

4244 

4289 

4334 

438o 

4o 

i9 

o325 

o37o 

o4i5 

o459 

o5o4 

o548 

4o 

20 

9.834425 

4470 

45i5 

456i 

46o6 

465i 

39 

20 

9.85o593 

o638 

0682 

°727 

0772 

0816 

39 

2  I 

4696 

474i 

4787 

4832 

4877 

4922 

38 

21 

0861 

o9o5 

o95o 

o995 

io39 

1084 

38 

22 

4967 

5oi2 

5o58 

5io3 

5i48 

5i93 

37 

22 

II29 

u73 

1218 

1262 

i3o7 

i352 

37 

23 

5238 

5284 

5329 

5374 

5419 

5464 

36 

23 

i396 

i44i 

i485 

i53o 

i575 

i6i9 

36 

24 

55o9 

5555 

56oo 

5645 

56oo 

5735 

35 

24 

1  664 

I7o8 

i753 

i797 

1842 

1887 

35 

25 

5780 

5826 

5871 

59i6 

59Gi 

6006 

34 

25 

i93i 

i976 

2O20 

2o65 

2110 

2i54 

34 

26 

6o5i 

6096 

6142 

6187 

6232 

6277 

33 

26 

2i99 

2243 

2288 

2332 

2377 

2422 

33 

27 

6322 

6367 

64  1  2 

6458 

65o3 

6548 

32 

27 

2466 

25ll 

2555 

2600 

2644 

2689 

32 

28 

6593 

6638 

6683 

6728 

6773 

68i9 

3i 

28 

2733 

2778 

2823 

2867 

29I2 

2956 

3i 

29 

6864 

6909 

6954 

6999 

7o44 

7o89 

3o 

2  9 

3ooi 

3o45 

3o9o 

3i34 

3i79 

3223 

3o 

3o 

9.837i34 

7179 

7225 

7270 

73i5 

736o 

29 

3o 

9.853268 

33i3 

3357 

3402 

3446 

349i 

29 

3i 

74o5 

745o 

7495 

754o 

7585 

763o 

28 

3i 

3535 

358o 

3624 

3669 

37i3 

3758 

28 

32 

33 

7675 
7946 

7721 
7991 

7766 
8o36 

781  1  7856 
8o8i|8i26 

7901 
8171 

27 
26 

32 

33 

38o2 
4o69 

3847 
4n4 

389i 
4i58 

3936 

4203 

3980 

4247 

4o25 

27 
26 

34 

8216 

8261 

83o7 

8352  8397 

8442 

25 

34 

4336 

438i 

4425 

4470 

45i4 

4559 

25 

35 

8487 

8532 

8577 

86228667 

8712 

24 

35 

46o3 

4648 

4692 

4737 

4781 

4826 

24 

36 

8757 

8802 

8847 

8892 

8937 

898a 

23 

36 

487o 

49i5 

4959 

5oo4 

5o48 

5o93 

23 

37 

9027 

9072 

9117 

9162 

9207 

9252 

22 

37 

5i37 

5i82 

5226 

527I 

53i5 

536o 

22 

38 

9297 

9343 

9388 

9433 

9478 

9523 

21 

38 

54o4 

5449 

5493 

5537 

5582 

5626 

21 

39 

9568 

9613 

9658 

97o3 

9748 

9793 

2O 

39 

5671 

57i5 

576o 

58o4 

5849 

5893 

2O 

4o 

9.839838 

9883 

9928 

9973 

..18 

..63 

I9 

4o 

9.855938 

5982 

6026 

6o7i 

6n5 

6160 

I9 

4i 

9.840108 

oi53 

oi98 

0243 

0288 

o333 

18 

4i 

6204 

6249 

6293 

6338 

6382 

6426 

18 

42 

o378 

0423 

o468 

o5i3 

o558 

o6o3 

I7 

42 

6471 

65i5 

656o 

66o4 

6649 

6693 

17 

43 

o648 

o693 

o737 

0782 

082-7 

0872 

16 

43 

6737 

6782 

6826 

687i 

69i5 

6959 

16 

44 

0917 

0962 

I007 

IO52 

io97 

1142 

i5 

44 

7004 

7o48 

7093 

7i37 

7l82 

7226 

i5 

45 

1187 

1232 

I277 

1322 

i367 

l4l2 

i4 

45 

7270 

73i5 

7359 

74o4 

7448 

7492 

i4 

46 

i457 

l5o2 

I  547 

l592 

i637 

1682 

i3 

46 

7537 

758i 

•7626 

767o 

77i4 

7759 

i3 

47 

1727 

1771 

1816 

1861 

I9o6 

i95i 

12 

47 

78o3 

7848 

7892 

7936 

7981 

8025 

12 

48 

1996 

2041 

2086 

2l3l 

2I76 

2221 

II 

48 

8o69 

8n4 

8i58 

8203 

824-7 

829i 

II 

49 

2266 

23ll 

2355 

240O 

2445 

2490 

10 

49 

8336 

838o8424 

8469 

85i3 

8558 

10 

5o 

9-842535 

258o 

2625 

2670 

27l5 

2760 

9 

5o 

9.  8586o2 

8646 

869i 

8735 

8779 

8824 

9 

5i 

28o5 

2849 

2894 

2939 

2984 

3o29 

8 

5i 

8868 

89I2 

8957 

9001 

9045 

9o9o 

8 

52 

3o74 

3n9 

3i64 

32O9 

3253 

3298 

7 

52 

9i34 

9i78 

9223 

926-7 

9811 

9356 

7 

53 

54 

3343 
36i2 

3388 
3657 

3433 

3702 

3478 

3747 

3523 
3792 

3568 
3837 

6 
5 

53 

54 

94oo  9444  9489 
9666  97io9755 

9533 
9799 

9577 
9843 

9622 

9888 

5 
5 

55 

3882 

3927 

397i 

4oi6 

4o6i 

4io6 

4 

55 

99352.9976  ..a  i 

..65 

.  109  .  i54 

4 

56 

4i5i 

4196  424r 

4285 

433o 

4375 

3 

56 

9.860198  0242  O287 

o33i 

o375  0420 

3 

57 

4420 

4465|45io 

4554 

4599 

4644 

2 

57 

o464  o5o8  o552  o597 

o64i  o685 

2 

58 

4689 

47344779 

4823 

4868 

49i3 

I 

58 

073007740818 

0862  0907^961 

I 

59 

4958 

5oo3  5o48 

5o92 

5i37 

5i82 

O 

Sg 

o995  io4o  io84 

1128  |II72  I2I7 

O 

GO"    |  50"   40" 

30'   20" 

10" 

g 

60"    |  50"   40"   30"  i  20"   10" 

a 

Co-tangent  of  55  Degrees. 

Co-tangent  of  54  Degrees. 

i 

pp   <  1"  2"  3"  4"  5"  6"  7"  8"  9" 

,  (  1"  2"  3''  4"  5"  6"  7"  8"  9" 

iri\  5   9  14  18  23  27  32  36  41 

/  4   9  13  18  C2  27  31  3(5  10 

GO 


LOGARITHMIC    SINES?. 


1  jt  |      Sine  of  36  Degrees. 

_g 

Sine  of  37  Degrees. 

SJ    0"    |  10"  |  20" 

30' 

40"   50" 

s 

0"    |  10" 

20" 

30" 

40" 

50" 

0 

i 

9.7692:9 
9393 

9248)9277  93o6 
942i  945o  9479 

9335936459 

95089537  58 

0 

I 

9-  779463 
963i 

949i 
9659 

9619 
9686 

9547 
97i4 

9575 

9742 

96o3 
977° 

59 
58 

2 

9566  9595 

96249653 

9682 

97n 

57 

2 

979s 

9826 

9854 

9882 

9910 

9938 

57 

3 

97409769 

97989827 

9856 

9884 

56 

3 

9966 

9998 

.  .21 

.149 

••77 

.io5 

56 

4    99189942 

997i 

.  .  .  . 

..29 

..58 

55 

4 

9.780188 

0161 

oi89 

0216 

0244 

0272 

55 

6:9.770087 

0116 

oi45 

oi73 

02O2 

023l 

54 

5 

o3oo 

0828 

o356 

o384 

o4n 

o439 

54  I 

6 

0260 

0289 

0818 

o347 

o376 

o4o4 

53 

6 

0467 

0495  o523 

o55i 

o578 

0606 

53 

7 

o433 

0462 

o49i 

O520 

o549 

o577 

52 

7 

o634 

0662  0690 

0-718 

o745 

o773 

52 

8 

0606 

o635 

o664 

o693 

0722 

o75o 

5i 

8 

0801 

0829 

o857 

0884 

0912 

o94o 

5i 

9 

°779 

0808 

08870866 

o895 

0928 

5o 

9 

0968 

0996 

1023 

io5i 

io79 

1107 

5o 

10 

9.77o952 

o98i 

IOIO 

io39 

io67 

1096 

49 

10 

9.  781184 

1162 

II9O 

1218 

1246 

1278 

49 

ii 

1125 

n54 

n83 

I2II 

1240 

1269 

48 

ii 

1801 

1829 

i357 

1  384 

l4l2 

i44o 

48 

12 

1298 

1826 

i355 

i384 

i4i3 

i44i 

47 

12 

i468 

i495 

i523 

i55i 

i578 

1606 

47 

i3    1470 

i499 

1628 

i556 

i585 

i6i4 

46 

18 

1  634 

1662 

i689 

i7i7 

i745 

1772 

46 

i4 

1  643 

1671 

1700 

1729 

i758 

i786 

45 

i4 

1800 

1828 

i856 

1888 

i9n 

I939 

45 

i5 

i8i5 

1  844 

1872 

1901 

1980 

i959 

44 

i5 

i966 

i994 

2O22 

2049 

2077 

2IO5 

44 

16 

i987 

2016 

2o45 

2073 

2IO2 

2181 

43 

16 

2182 

2160 

2188 

22l5 

2243 

2271 

43 

17 

2188 

2217 

2245 

2274 

23o3 

42 

17 

2298 

2826 

2354 

2881 

24o9 

2437 

42 

18 

233i 

2860 

2389 

24l7 

2446 

2475 

4i 

18 

2464 

2492 

252O 

2547 

2575 

2602 

4i 

Z9 

25o3 

2532 

256i 

2589 

2618 

2646 

4o 

i9 

2680 

2658 

2685 

2-718 

274l 

2768 

4o 

209.772675 

2704 

2782 

276l 

2790 

2818 

39 

20 

9.782796 

2823 

285i 

2879 

29o6 

2934 

39 

21 

2847 

2875 

2904 

2933 

296l 

2990 

38 

21 

2961 

2989 

3oi7 

3o44 

3o72 

3o99 

38 

22 

3oi8 

8047 

3o76 

3io4 

3i33 

3i6i 

37 

22 

8127 

3i54 

8182 

32IO 

3237 

5265 

37 

23 

8190 

32I9 

8247 

3276 

33o4 

3333 

36 

23 

3292 

3320 

3347 

3375 

3402 

343o 

36 

24 

336i 

339o 

34i8 

3447 

3476 

35o4 

35 

24 

3458 

3485 

35:3 

354o 

3568 

3595 

35 

25 

3533 

356i 

359o 

36:8 

3647 

3675 

34 

25 

3623 

365o 

3678 

37o5 

3733 

376o 

34 

26 

37o4 

3732 

376i 

3789 

38i8 

3846 

33 

26 

3788 

38i5 

3843 

387o 

3898 

3925 

33 

27 

3875 

39o3 

3932 

396o 

3989 

4oi7 

32 

27 

3953 

8980 

4oo8 

4o35 

4o63 

4o9o 

82 

28 

4o46 

4074 

4io34i3i 

4i6o 

4i88 

3i 

28 

4n8 

4i45 

4i73 

4200 

4228 

4255 

3i 

29 

421-7 

4245 

42744302 

433i 

4359 

3o 

29 

4282 

4810 

4337 

4365 

4392 

4420 

3o 

3o 

g.774388 

44i6 

4445  4473 

45oi 

453o 

29 

3o 

9.  784447 

4475 

45o2 

4529 

4557 

4584 

29 

3i 

32 

4558 
4729 

4587 
4757 

46i54644 
47864814 

4672 

4842 

47oo 

487i 

28 
27 

3i 
82 

4612 

4776 

4639 
48o4 

4667 
483i 

4694 

4858 

472I 

4886 

49i3 

28 
37 

33 

4899 

4928 

49564985 

5oi3 

5o4i 

26 

33 

494i 

4968 

4995 

5o23 

5o5o 

5o78 

26 

34 

5o7o 

5o98 

5i26 

5i55 

5i83 

5212 

25 

34 

5io5 

5i32 

5i87 

52i4 

5242 

25 

35 

6240 

5268 

5297 

5325 

5353 

5382 

24 

35 

5269 

5296 

5324 

535i 

5378 

54o6 

24 

36 

54io 

5438 

5467 

5495 

5523 

5552 

23 

36 

5433 

546i 

5488 

55i5 

5543 

557o 

23 

37 

558o 

56o8 

56375665 

5693 

5722 

22 

37 

5597 

5624 

5652 

5679 

57o6 

5734 

22 

38 

575o 

5778 

58o7 

5835 

5863 

5892 

21 

38 

576i 

5788 

58i6 

5843 

587o 

5898 

21 

39 

592O 

5948 

5977 

6oo5 

6o33 

6061 

2O 

39 

5925 

5952 

5979 

6oo7 

6o34 

6061 

2O 

4o 
4i 

9.776o9o 
6269 

6118 

6288 

6i46  6i75i62o3 
63i663446372 

6281 
6401 

18 

4o 
4i 

9.786o89 

6252 

6116 
62-79 

6i43 
63o7 

6i7o 
6334 

6i98 
6861 

6225 

6388 

18 

42 

6429 

6457 

6485 

65i46542 

6570 

17 

42 

64i6 

6443 

647o 

6497 

6525 

6552 

1-7 

43 

6598 

6627 

6655 

6683  67n 

6789 

16 

43 

6579 

6606 

6634 

6661 

6688 

67i5 

16 

44 

6768 

6796 

6824 

6852  6880 

69o9 

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44 

6742 

677o 

6797 

6824 

685i 

6878 

i5 

45 

6937 

6965 

6993 

7O2I 

7o5o 

7078 

i4 

45 

69o6 

6933 

696o 

6987 

7oi4 

7042 

i4 

46 

7106 

7i34 

7162 

-7191 

72I9 

7247 

18 

46 

7o69 

7o96 

7I23 

7i5o 

7i77 

7205 

i3 

47 

7275 

7808 

733i 

7359 

7388 

74i6 

12 

47 

7232 

7259 

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73i3 

734o 

7367 

12 

48 

7444 

7472 

75oo 

7528 

7556 

7585 

II 

48 

7395 

7422 

7449 

7476 

75o3 

753o 

I  I 

49 

76i3 

7641 

7669 

7697 

7725 

7753 

10 

49 

7557 

7585 

76l2 

7639 

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7693 

10 

5o 

9-77778i 

7810 

7838 

7866 

7894 

7922 

9 

5o 

9.78772o 

7747 

7774 

-7801 

7829 

7856 

9 

5i 

795o 

7978 

8006 

8o34 

8062 

8o9i 

8 

5i 

7883 

-7910 

7937 

7964 

7991 

8018 

8 

62    8119 

8i47 

8r75 

8208 

8281 

8269 

7 

52 

8o45 

80-72 

8o99 

812-7 

8i54 

8181 

7 

53    8287 

8816 

8343 

837i 

8399 

8427 

6 

53 

8208 

8235 

8262 

8289 

8816 

8343 

6 

54    8455 

8483 

85n 

85398567 

8595 

5 

54 

837o 

8397 

8424 

845  1 

8478 

85o5 

5 

55 
50 
57 

8624 
8792 
896o 

8652 
8820 
8988 

8680 
8848 
9oi6 

87o8  8736 
88-768904 
9044  9072 

8764 
8932 
9ioo 

4 
3 

2 

55 
56 
57 

8532 
8694 
8856 

86698686 

8721  8748 
8883  89io 

86i3864o 

877588o2 

8667 
8829 
899i 

4 
3 

2 

58 

9.128 

9i56 

9i83 

9211 

9289 

9267 

I 

Co 
Do 

9oi8 

9o45  9o72 

9o99  9i26 

91  53 

I 

59 

9295 

9323 

935i 

93-79  940-7 

9435 

0 

59 

9i8o  92o7  9234 

9261^9288  93i5 

O 

60"     50" 

40" 

30"   20" 

10" 

S 

60"     50"   40"  |  30"   20"   10"  |  ^ 

Co-sine  of  53 

Degrees. 

Co-sine  of  52  Degrees. 

<  1"  2"  3"  4" 

5"  6"  7"  8"  9" 

t  it/  o"  •}/'  4"  V  fi"  7"  8"  Q"  ' 
P  P-irt  >1~d4')0'8 

r.PartJ  3   6   9  11 

14  17  20  23  26 

i 

i  .  r  an  <  ty   K   o   -.i   -11  if.  iQ  ^Q  or- 
£  <j   o   o   11  14  lo  iy  ~z  £•) 

....  .,  .._,      .  ,     .  —  ,.  .  . 

LOGARITHMIC    TANGENTS. 


61 


jj 

Tangent  of  36  Degrees. 

.s 

Tangent  of  37  Degrees. 

s 

o» 

10" 

20" 

30" 

40" 

50" 

s 

0" 

10" 

20" 

30"   40" 

50" 

oj9.  861261 

i3o5 

i35o 

i394 

i438 

1482 

59 

o 

9.877114 

7i58 

72O2 

7246  7290 

7333 

59 

ij      l527 

i57i 

i6i5 

i659 

i7o4 

1748 

58 

i 

7377 

7421 

7465 

75o9 

7552 

7696  58 

2       I792 

i837 

1881 

I925 

1969 

20l4 

57 

2 

7684 

7728 

7771 

78i5 

7859  57 

3;    2o58 

2102 

2146 

2191 

2235 

2279 

56 

3 

79°3 

7947 

799° 

8o34 

8o78 

8122  56 

4!    2323 

2368 

2412 

2456 

2500 

2545 

55 

4 

8i65 

8209 

8253 

8297 

834i 

8384 

55 

5    2689 

2633 

2677 

2721 

2766 

2810 

54 

5 

8428 

8472 

85i6 

855o 

86o3 

8647 

54 

6 

2854 

2898 

2943 

2987 

3o3i 

3075 

53 

6 

8691 

8734 

8778 

8822 

8866 

8909 

53 

7 

3u9 

3i64 

3  2  08 

3252 

3296 

334i 

52 

7 

8963 

8997 

9o4i 

9o85 

9128 

917-2 

52 

81    3385 

3429 

3473 

35i7 

3562 

36o6 

5i 

8 

9216 

9260 

93o3 

9347 

939i 

9435 

5i 

9    365o 

3694 

3738  3783 

3827 

387i 

5o 

9 

9478 

9522 

9566 

9609 

9653 

9697 

5o 

10 

1  1 

9.860915 

4  1  80 

3959 

4225 

4269  43  1  3 

4092 

4357 

4i.36 
44oi 

49 

48 

10 
ii 

9.87974i 
9.88ooo3 

9784 
0047 

9828 
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9872 
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9916 
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9959 

0222 

49 

48 

12 

4445 

449o 

4534 

4578 

4622 

4666 

47 

12 

0265 

0309 

o353 

o397 

o44o 

o484 

47 

i3 

47io 

4755 

4799 

4843 

4887 

493i 

46 

i3 

o528 

0571 

o6i5 

o659 

o7o3 

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46 

i4 

4975 

5020 

5o64 

5io8 

5i52 

5i96 

45 

i4 

0-790 

o834 

0877 

0921 

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1008 

45 

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5285 

5329 

5373 

54i7 

546i 

44 

i5 

IO52 

1096 

n4o 

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1271 

44 

16 

55o5 

5549 

5594 

5638 

5682 

5726 

43 

16 

i3i4 

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1402 

i445 

i489 

i533 

43 

17 

577o 

58i4 

5858 

59o3 

5947 

599i 

42 

17 

i577 

1620 

1  664 

1-708 

i75i 

1795 

42 

18 

6o35 

6o79 

6i23 

6i67 

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8256 

4i 

18 

1839 

1882 

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i97o 

20l3 

2057 

4i 

19 

63oo 

6344 

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6432 

6476 

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4o 

19 

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2i44 

2188 

2232 

2275 

2319 

4o 

20 

9.  866564 

66o9 

6653 

6697 

674i 

6785 

39 

20 

9.882363 

24062450 

2494 

2537 

258i 

39 

21 

6829 

6873 

69i7 

696i 

•7006 

7o5o 

38 

21 

2625 

2668  2712 

2756 

2799 

2843 

38 

22 

7o94 

7i38 

7182 

7226 

7270 

73i4 

37 

22 

2887 

293o 

2974 

3oi8 

3o6i 

3io5 

3? 

23 

7358 

7402 

7446 

749i 

7535 

7579 

36 

23 

3i48 

3192 

3236 

3279 

3323 

3367 

36 

24 

7623 

7667 

7711 

7755 

7799 

7843 

35 

24 

34io 

3454 

3498 

354i 

3585 

3628 

35 

25 

7887 

793i 

7975 

8oi9 

8o64 

8108 

34 

25 

3672 

37i6 

3759 

38o3 

3847 

3890 

M 

26 

8i52 

8i96 

8240 

8284 

8328 

8372 

33 

26 

3934 

3977 

4021 

4o65 

4io8 

4i52 

33 

27 

84i6 

846o 

85o4 

8548 

8592 

8636 

32 

27 

4196 

4283 

4326 

437o 

44i4 

32 

28 

8680 

8724 

8768 

88i3 

8857 

89oi 

3i 

28 

4457 

45oi 

4544 

4588 

4632 

4675 

3i 

29 

8945 

8989 

9o33 

9°77 

9121 

9i65 

3o 

29 

4-719 

4762 

48o6 

485o 

4893 

4937 

3o 

3o 

9.8692O9 

9253 

9297 

934i 

9385 

9429 

29 

3o 

9.884980 

5o24 

5o68 

5m 

5i55 

5i98 

29 

3i 

9473 

95i7 

956i 

96o5 

9649 

9693 

28 

3i 

5242 

5286 

5329 

5373 

54i6 

546o 

28 

32 

9737 

9781 

9825 

9869 

99i3 

9957 

27 

32 

55o4 

5547 

559i 

5634 

5678 

572I 

27 

33 

9.87oooi 

oo45 

0089 

oi33 

OI77 

O22I 

26 

33 

5765 

5809 

5852 

5896 

5939 

5983 

26 

34 

0265 

oSog 

o353 

o397 

o44i 

o485 

25 

34 

6026 

6o7o 

6114 

6i57 

6201 

6244 

25 

35 

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o573 

0617 

0661 

0705 

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24 

35 

6288 

633i 

6375 

6419 

6462 

65o6 

24 

36 

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o837 

0881 

0925 

0969 

ioi3 

23 

36 

6549 

6593 

6636 

6680 

6723 

6767 

23 

37 

io57 

IIOI 

n45 

1189 

1233 

1277 

22 

37 

6811 

6854 

6898 

694i 

6985 

7028 

22 

38 

l32I 

i365 

i4o9 

i453 

1497 

i54i 

21 

38 

7072 

7ix5 

7i59 

72O2 

7246 

7289 

21 

39 

i585 

1629 

i673 

1717 

1761 

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20 

39 

7333 

7377 

7420 

7464 

75o7 

755i 

20 

4o 

9.87i849 

i893 

i937 

1980 

2024 

2068 

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4o 

9.887594 

7638 

768i 

7726 

7768 

7812 

19 

4i 

2112 

2i56 

2200 

2244 

2288 

2332 

18 

4i 

7855 

7899 

7942 

7986 

8029 

8o73 

18 

42 

2376 

2420 

2464 

25o8 

2552 

2596 

17 

42 

8116 

8160 

8203 

824-7 

8291 

8334 

17 

43 

2640 

2684 

2727 

277I 

2816 

2859 

16 

43 

8378 

8421 

8465 

85o8 

8552 

8595 

16 

44 

2903 

2947 

299I 

3o35 

3o79 

3i23 

i5 

44 

8639 

8682 

8726 

8769 

88i3 

8856 

i5 

45 

3i67 

3211 

3255 

3299 

3342 

3386 

i4 

45 

8900 

8943 

8987 

9o3o 

9074 

9117 

i4 

46 

343o 

3474 

35i8 

3562 

36o6 

365o 

i3 

46 

9161 

9204 

9248 

9291 

9334 

9378 

i3 

47 

3694 

3738 

378i 

3825 

3869 

39i3 

12 

47 

9421 

9465 

95o8 

9552 

9595 

9639 

12 

48 

3957 

4ooi 

4o45 

4o89 

4i33 

4177 

II 

48 

9682 

9?26 

9769 

98i3 

9856 

99°° 

II 

49 

4220 

4264 

43o8 

4352 

4396 

444o 

10 

49 

9943 

9987 

..3o 

..74 

.117 

.  1  60 

10 

5o 

9,874484 

4528 

4572 

46i5 

4659 

47o3 

9 

5o 

9.890204 

0247 

O29I 

o334 

o378 

0421 

9 

5i 

4747 

4791 

4835 

4879 

4923 

4966 

8 

5i 

o465 

o5o8 

o552 

o595 

o639 

0682 

8 

52 

5oio 

5o54 

5o98 

5i42 

5i86 

523o 

7 

52 

0725 

o769 

0812 

o856 

o899 

o943 

7 

53 

52?3 

5317 

536i 

54o5 

5449 

5493 

6 

53 

0986 

io3o 

io73 

1116 

1  1  60 

1203 

6 

54 

5537 

558o 

5624 

5668 

5712 

5756 

5 

54 

1247 

I  200 

1  334 

i377 

1421 

1  464 

5 

55 

58oo 

5843 

5887 

593i 

5975 

6oi9 

4 

55 

1507 

1  55  1 

i594 

i638 

1681 

1725 

4 

56 

6o63 

6107 

6i5o 

6194 

6238 

6282 

3 

56 

1768 

1811 

i855 

1898 

I9/l2 

i985 

3 

57 

6326 

6370 

64i3 

6457 

65oi 

6545 

2 

57 

2028 

2O72 

2Il5 

2i59 

2202 

2246 

2 

58 

6589 

5632  6676 

6720 

6764 

6808 

I 

58 

2289 

2332 

2376 

2419  2463 

25o6 

I 

59 

6852 

6895  6939 

6983 

7027 

7071 

O 

^9 

2549 

25932636 

2680!2723 

2766 

O 

tiO"    |  50"  |  40" 

30" 

20" 

10" 

_g 

60"     50"   40" 

30"   20"  1  10" 

~T 

Co-tangent  of  53  Degrees. 

Co-tangent  of  52  Degrees. 

§ 

pp   C  1"  2"  3"  4"  5"  6"  7"  8"  9"  \\      .  $  1"  2"  3"  4"  5"  6"  7'  8"  9" 

r<lurt>  4   9  13  18  22  26  31  35  40  ||  J-lart{  4   9  13  17  22  2G  31  35  39 

LOGARITHMIC    SINES. 


n  [     Sine  of  38  Degrees. 

c 

Sine  of  39  Degrees. 

&  [~   0"    |  10" 

20"   30"   40" 

50" 

&     0"      10" 

20"  |  30"  |  40" 

50" 

0 
V 

9.789342  9369 
95o4  9  * 
9665  9692 

939619423  9450 
955-7195849611 
97i997469773 

9477 

9638 
9800 

59 

58 
57 

o 
I 

2 

9.798872  8898 
9028  9054 
918419210 

8924 
9080 
9236 

895o8976 
9106  9132 
9262  928-7 

9002 

9i58 
93i3 

59 
58 
57 

3 

9827  9804 

9880  99o7j9934  996i 

56 

3 

93399365 

939i 

94i7 

9443 

9469 

56 

4 
5 
6 

9988 
9  .790149 
o3io 

oi76 
o337 

..42 

0203 

o364 

|..69 

O230 

0391 

..96 

O257 

o4i8 

.  122 

0284 

o445 

55 
54 
53 

4 
5 
6 

9495  9521  9547 
9651196779703 
98o698329858 

9573 

9728 

9884 

9599 
97^4 
9910 

9625 
978o 
9936 

55 
54 
53 

7 

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1-768 

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4i 

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9.792557 

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9.8019-73 

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24 

3i95 

3222 

3248 

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35 

24 

2589 

26i5 

2641 

2666 

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27l8 

35 

25 

3354 

338i 

34o7 

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346o 

3487 

34 

25 

2743 

2769 

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2846 

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34 

26 

35i4 

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3620 

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2S97 

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29 

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9  .  794i5o 

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3o 

9.8o35n 

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3i 

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436i 

4388 

44i4 

444i 

28 

3i 

3664 

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37i5 

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3766 

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28 

32 

446  7 

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4546 

4573 

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27 

32 

38i7 

3842 

3868 

3893 

39i9 

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27 

33 

4626 

4652 

4678 

47o5 

473i 

4758 

26 

33 

3970 

3995 

4021 

4o46 

4o72 

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34 

4784 

48io4837 

4863 

489o 

4916 

25 

34 

4i23 

4i48 

4i74 

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35 

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5o48 

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35 

4276 

43oi 

4327 

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36 

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69i4 

694i 

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12 

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48 

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49 

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49 

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9.806557 

6583 

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6633 

6658 

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9 

5i 

7464 

749o; 

75i6 

7542 

7569 

7595 

8 

5i 

6709 

6734 

6759 

6785 

6810 

6835 

8 

52 

•7621 

7647 

7673 

7699 

7725 

775i 

7 

52 

6860 

6885 

6911 

6936 

6961 

6986 

7 

53 

7777 

78o4 

783o 

7856 

7882 

7908 

6 

53 

7OII 

7o37 

7062 

7o87 

7II2 

7i37 

6 

54 

7934 

796o 

7986 

8012 

8o38 

8o65 

5 

54 

7i63 

7188 

7213 

7238 

7263 

7288 

5 

55 

8091 

81178143 

8i69 

8195  8221 

4 

55 

73i4 

7339 

7364 

7389 

74i4 

7439 

4 

56 

8247 

8273,8299 

8325 

835i  8377 

3 

56 

7465 

749o 

75i5 

754o 

7565 

759° 

3 

57 

84o3 

8429!8455848i 

85o88534 

2 

57 

76i5 

7641 

7666 

-7691 

77i6774i 

2 

58 

856o 

858686128638 

8664j869o 

I 

58 

7766 

7791 

7816 

7842 

7867  7892 

I 

59 

87i6 

874287688794 

8820J8846 

O 

_5o_ 

79i7 

794s  7967 

-7992 

8oi7  8042 

O 

60" 

50" 

40"  |  30"   20"  |  10" 

S" 

60"     50"  |  40"  |  30"   20"   10" 

j 

Co-sine  of  51  Degrees. 

Co-sine  of  50  Degrees. 

i 

1.  T>art$  l"  Z"  3/ 

4"  5"  6"  7"  8"  9" 

(  1"  2"  3"  4"  5"  6"  7"  8"  9" 

1  rl\  358 

11  13  16  19  21  24  I 

trtj  3   5   8  10  13  15  18  20  23 

LOGARITHMIC    TANGENTS. 


g  |    Tangent  of  38  Degrees. 

d 

Tangent  of  39  Degrees. 

1 

SJ    0"    ,  10" 

20" 

30'' 

40" 

50" 

* 

0"    |  10"   20"   30"  |  40"   50" 

019.892810 

2897 

2940 

2983 

3o27 

59 

0 

9.9o8369|84i2 

8455 

8498 

854i 

8584 

59 

i 

3o7o 

3n4 

3i57 

320O 

3244 

3a87 

58 

i 

86288671 

8714 

8757 

8800 

8843 

58 

2 

333i 

3374 

34i7 

3461 

35o4 

3547 

57 

2 

8886 

8929 

8972 

9oi5 

9o58 

O.IOI 

57 

3 

359i 

3634 

3678 

372I 

3764 

38o8 

56 

3 

9i44 

9187 

9230 

9273 

93i6 

9359 

56 

4 

385i 

3894 

3938 

398i 

4o25 

4o68 

55 

4 

9402 

9445 

9488 

953i 

9574 

96i7 

55 

5 

4ni 

4i55 

4198 

4241 

4285 

4328 

54 

5 

966o 

97o3 

9746 

9789 

9832 

9875 

54 

6 

4372 

44i5 

4458 

45o2 

4545 

4588 

53 

6 

99i8 

9961 

...5 

..48 

.  .9i 

.i34 

53 

7 

4632 

4675 

4718 

4762 

48o5 

4848 

52 

7 

9.9ioi77 

O220 

0263 

o3o6 

o349 

o392 

52 

8 

4892 

4935 

4979 

5022 

5o65 

5io9 

5i 

8 

o435 

o478 

0521 

o564 

060-7 

o65o 

5i 

9 

5i52 

5i95 

5239 

5282 

5325 

5369 

5o 

9 

o693 

o736 

°779 

0822 

o865 

o9o8 

5o 

IO 

9.8954i2 

5455 

5499  5542 

5585 

5629 

49 

IO 

9.910951 

0994 

1037 

1080 

1123 

1166 

t 

ii 

5672 

57i5 

5759  58o2 

5845 

5889 

48 

ii 

I2O9 

1262 

I295 

i338 

i38i 

1424 

48 

12 

5932 

5975 

601916062 

6io5 

6i49 

47 

12 

i467 

i5io 

i553 

i596 

i639 

1682 

47 

i3 

6l92 

6235 

6278 

63226365 

64o8 

46 

i3 

I725 

1768 

1810 

i853 

i896 

i939 

46 

i4 

6452 

6495 

6538 

6582 

6625 

6668 

45 

i4 

I982 

2O25 

2068 

21  II 

2  1  54 

2197 

45 

i5 

6712 

6755 

6798 

6842 

6885 

60.28 

44 

i5 

2240 

2283 

2326 

2369  24l2 

2455 

44 

16    6971 

7oi5 

7o58 

7IOI 

7i45 

7i88 

43 

16 

2498 

254i 

2584 

2627  2670 

27i3 

43 

17 

7231 

7275 

73i8 

736i 

74o4 

7448 

42 

17 

2756 

2799 

2842 

2&85!2928 

29-71 

42 

18 

7491 

7534 

7578 

-7621 

7664 

77°7 

4i 

18 

3oi4 

3o57 

3ioo 

3i43j3i85 

3228 

4i 

i9 

775i 

7794 

7837 

788i 

-7924 

7967 

4o 

i9 

327I 

33i4 

3357 

34oo  3443 

3486 

4o 

20 

9.898oio 

8o54 

8o97 

8i4o 

8:83 

8227 

39 

20 

9.9i3529 

3572 

36i5 

3658  37oi 

3744 

39 

21 

8270 

83i3 

8357 

84oo 

8443 

8486 

38 

21 

3787 

383o 

3873 

39i6 

3959 

4ooi 

38 

22 

853o 

8573 

8616 

8659 

87o3 

8746 

37 

22 

4o44 

4o87 

4i3o 

4i73 

4216 

4259 

37 

23 

8789 

8832 

8876 

89i9 

8962 

9oo5 

36 

23 

4302 

4345 

4388 

443  1 

4474 

45i7 

36 

24 

9049 

9092 

9i35 

9i78 

9222 

9265 

35 

24 

456o 

46o3 

4645 

4688 

473i 

4774 

35 

25 

93o8 

935i 

9395 

9438 

948i 

9524 

34 

25 

48i7 

486o 

49o3 

4946 

4989 

5o32 

34 

26 

9568 

96n 

9654 

9697 

974i 

9784 

33 

26 

5o75 

5n8 

5i6i 

52o3 

5246 

5289 

33 

27 

0827 

0870 

QQl4 

0,0,57 

43 

32 

27 

5332 

5375 

54i8 

546i 

55o4 

5547 

32 

7  / 
28 

v  / 
9.90008-7 

7   / 

oi3o 

yy*** 

oi73 

77   / 
02l6 

0259 

o3o3 

3i 

—  / 
28 

559o 

^/^i  i  -j 
5633 

5675 

57i8 

576i 

*  ***»  / 

58o4 

3i 

29 

o346 

o389 

0432 

o476 

o5i9 

o562 

3o 

29 

5847 

5890 

5933 

5976 

6019 

6062 

3o 

3o 

9.900605 

o648 

o692 

o735 

o778 

0821 

29 

3o 

9.916104 

6147 

6i9o 

6233 

62-76 

63i9 

29 

3i 

0864 

o9o8 

o95i 

o994 

io37 

1081 

28 

3i 

6362 

64o5 

6448 

649i 

6533 

6576 

2$ 

32 

1124 

1x67 

1210 

1253 

I297 

i34o 

27 

32 

66i9 

6662 

67o5 

6748 

679i 

6834 

27 

33 

i383 

1426 

i469 

i5i3 

i556 

i599 

26 

33 

6877 

6919 

6962 

7oo5 

7o48 

7o9i 

26 

34 

1642 

i685 

I729 

I772 

i8i5 

i858 

25 

34 

7177 

-7220 

-7262 

73o5 

7348 

25 

35 

1901 

i944 

i988 

2031 

2074 

2II7 

24 

35 

739i 

7434 

7477 

752O 

7563 

76o5 

24 

36 

2160 

2204 

2247 

229O 

2333 

2376 

23 

36 

7648 

7691 

7734 

7777 

-7820 

7863 

23 

37 

2420 

2463 

25o6 

2549 

2592 

2635 

22 

37 

79o6 

7948 

7991 

8o34 

8o77 

8120 

22 

38 

2679 

2-722 

2765 

2808 

285i 

2894 

21 

38 

8i63 

8.206 

8248 

829I 

8334 

8377 

21 

39 

2938 

298i 

3024 

3o67 

3no 

3i53 

20 

39 

8420 

8463 

85o6 

8548 

859i 

8634 

2O 

4o 

9.9o3i97 

324o 

3283 

3326 

3369 

34i2 

I9 

4o 

9.918677 

8720 

8763 

88o5 

8848 

889i 

I9 

4i 

3456 

3499 

3542 

3585 

3628 

367i 

18 

4i 

8934 

8977 

9O2O 

9o63 

9io5 

9i48 

18 

42 

37i4 

3758 

38oi 

3844 

3887 

393o 

17 

42 

9191 

9234 

9277 

932O 

9362 

94o5 

i7 

43 

3973 

4oi6 

4o6o 

4io3 

4i46 

4i89 

16 

43 

9448 

9491 

9534 

9577 

9619 

9662 

16 

44 

4232 

4275 

43i8 

4362 

44o5 

4448 

i5 

44 

9705 

9748 

979i 

9834 

9876 

99i9 

i5 

45 

449i 

4534 

4577 

4620 

4663 

4707 

i4 

45 

9962 

...5 

.  48 

.  .9i 

.i33 

.176 

i4 

46 

475o 

4793 

4836 

4879 

4922 

4965 

i3 

46 

9.920219 

0262 

o3o5 

o348 

o39o 

o433 

i3 

47 

5oo8 

5o52 

5o95 

5i8i 

5224 

12 

47 

0476 

oSig 

o562 

0604 

o647 

o69o 

12 

48 

5267 

53io 

5353 

5397 

544o 

5483 

II 

48 

o733 

0776 

08  1  9 

0861 

o9o4 

o947 

II 

49 

5526 

5569 

56i2 

5655 

5698 

574i 

IO 

49 

0990 

io33 

io75 

1118 

1  161 

1204 

10 

5o 

9.9o5785 

5828 

587i 

59i4 

5957 

6000 

9 

5o 

9.921247 

1289 

i332 

i375 

i4i8 

1  46  1 

9 

5i 

6o43 

6086 

6129 

61-72 

6216 

6259 

8 

5i 

i5o3 

1  546 

i589 

i632 

i675 

1717 

8 

52 

63o2 

6345 

6388 

643  1 

6474 

65  1  7 

7 

52 

1760 

i8o3 

1  846 

i889 

i93i 

19-74 

7 

53 

656o 

66o3 

6646 

669o 

6733 

6776 

6 

53 

2017 

2060 

2103 

2145 

2188 

223l 

6 

54 

68i9 

6862 

6905 

6948 

699i 

7034 

5 

54 

2274 

23i6 

2359 

2402 

2445 

2488 

5 

55 

7o77 

7120 

7i63 

7207 

725o 

7293 

4 

55 

253o 

2573 

2616 

2659 

27O2 

2744 

4 

56 

7336 

7379 

7422 

7465 

75o8 

755i 

3 

56 

2787 

283o 

2873 

29l5 

2958 

3ooi 

3 

57 

7594 

7637 

768o 

7723 

7766 

78o9 

2 

57 

3o44 

3o87 

3i29 

3l72 

32i5 

3258 

2 

58 

7853 

7896 

7939 

7982 

8025 

8068 

I 

58 

33oo 

3343 

3386 

3429 

347i 

35i4 

I 

59 

8xi  i 

8i54 

8i97 

8240 

8283 

8326 

o 

59 

3557 

36oo 

3642 

3685 

3728 

377i 

0 

60' 

50"  |  40" 

30" 

20-- 

10" 

a* 

60"     50" 

40" 

30" 

20" 

10" 

t! 

Co-tangent  of  5  1  Degrees. 

Co-tangent  of  50  Degrees. 

2 

p  r  t  (  1"  2"  3"  4"  5"  6"  7"  8"  9" 

„  v   C  I"  2"  3"  4"  5"  6"  7"  8"  9" 

irl\  i   9  13  17  22  26  30  35  39 

I.Fart^  4   9  13  17  01  26  30  34  39 

LOGARITHMIC    SINES. 


d 

Sine  of  40  Degrees. 

o 

Sine  of  41  Degrees. 

§ 

0" 

10" 

20"  i  30"   40" 

50" 

s 

0" 

10" 

20" 

30" 

40" 

50" 

0 

o  .  8o8o67 

8o93 

8n88i438i68 

8i93 

59 

O 

9.8i6943 

6967 

699i 

7016 

7040 

7o64 

59 

I 

8218 

8243 

826882938318 

8343 

58 

I 

7088 

71  12 

7i37 

7161 

7i85 

-7209 

58 

2 

8368 

8393 

841984448469 

8494 

57 

2 

7233 

7258 

-7282 

73o6 

733o 

7354 

57 

3 

85i9 

8544 

8569;8594;86i9 

8644 

56 

3 

7379 

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9.811061 

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0 

59 

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0 

60"    |  50"  | 

40"  |  30" 

20"   10" 

.9 

60" 

50" 

40" 

30" 

20" 

10" 

d 

Co-sine  of  49  Degrees. 

Co-sine  of  48  Degrees. 

3 

p  P  .<  1"  2"  3"  4"  5"  6"  7"  8"  9" 

.  (  1"  2"  3"  4"  5"  6"  7"  8"  9" 

l.lartj  257 

10  12  15  17  20  22 

in\  2  5   7  10  12  14  17  19  2t 

L  o  G  A  R  i  T  ii  M  i  c    TANGENTS. 


Jj 

Tangent  of  40  Degrees. 

.5 
?, 

Tangent'  of  4  L 

Degrees. 

30"  |  40"  |  50"  t 

0" 

10" 

20" 

30" 

40"  |  50" 

0" 

10"   20" 

0 

9,923814 

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8 

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8 

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9 

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1  544 

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12 

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28 

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33 

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2394 

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33 

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52 

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53 

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5 

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4 

55 

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3 

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2 

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O 

59 

4i83;4226  4268.4310  43534395 

0 

h60"     50"   40" 

30"  |  20"  |  10" 

a 

60"     50"   40"  |  30"   20'   .10" 

£3 

Co-tangent  of  49  Degrees. 

•9 

& 

Co-tangent  of  48  Degrees. 

1' 

p  p   <  1"  2"  3"  4"  5"  6;'  7"  8"  9" 
j  4   9  13  17  21  26  30  34  38 

p  p  fC  1"  2"  3"  4"  5"  6''  7'  8"  9" 
irt}  4   8  13  17  21  25  30  34  38 

E 


GG 


LOGARITHMIC    SINES. 


jf 

Sine  of  42  Degrees. 

d 

Sine  of  43  Degrees. 

m 

0" 

10" 

20"  |  30" 

40"   50" 

s 

0" 

10"   20"  |  30" 

4(X'  1  50" 

o 

9.$255u 

5534 

5558 

558i 

56o4  5628 

59 

0 

9,833783 

38o6 

3828 

385i 

3874'38q6 

59 

2 

565i  5675 
579i  58i5 

5698  572i 
5838586i 

5745 
5885 

576858 
590867 

i 

39i9 
Ao54 

394i 
4077 

3964  398614009 

4o99  4122  4i44 

4o33 

4167 

53 
57 

3 

593i  5955 

5978 

6001 

6025  6o48 

56 

3 

4i89 

42T2 

4234  4257 

4280 

4302 

56 

4 

6071  6o95 

6118 

6i4i  6i65|6i88 

55 

4 

4325 

4347  437o  4392 

44  1  5 

4437 

55 

5 

6211 

6235 

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54 

5 

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4527 

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4572 

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6 

635i 

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53 

6 

4595 

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4662 

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53 

7 

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65i4 

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52 

7 

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48 

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48 

12 

7i89 

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7235 

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47 

12 

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5493 

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7629 

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44 

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5829 

585i 

5874 

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44 

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7745 

7768 

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43 

16 

594i 

5963 

5986 

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43 

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7884 

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793i 

7954 

7977  8000 

42 

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6o75 

6o97 

6120 

6142 

6i64 

6187 

42 

18 

8o23 

8o46 

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18 

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623i 

6254 

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632i 

4i 

19 

20 
21 

8162 

q.  828301 

8439 

8i85 
8324 
8462 

82088231  825482-78 
8347837o839384i6 
8485  85o9  8532  8555 

4o 
39 

38 

'9 

20 
21 

6343 
o.S36477 
6611 

6365 
6499 
6633 

6388 

6522 

6656 

64io 
6544 
6678 

6432 
6566 
67oo 

6455 
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6722 

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39 
38 

22      8578 

8601 

8624  8647 

8670 

8693 

37 

22 

6745 

6767 

678q 

6812 

6834 

6856 

37 

23      87l6 

8739 

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88o9 

8832 

36 

23 

6878 

69oi 

6923 

6945 

6968 

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36 

24    8855 

8878 

89oi 

8924 

8947 

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35 

24 

7OI2 

7034 

7o57 

7079 

7IOI 

7I23 

35 

25i    8993 

9Ol6 

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9o85 

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34 

25 

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7i68 

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26 

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33 

26 

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28 

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28 

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29 

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29 

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7746 

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29 

3o 

9.8378i2 

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29 

3i 

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99i3 

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28 

3i 

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28 

32 

9959 

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27 

32 

8078 

8100 

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8i45 

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27 

33  9.83000.7 

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0142 

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0188 

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26 

33 

8211 

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26 

34!    0234 

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25 

34 

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25 

35    0372 

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24 

35 

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852i 

8543 

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36 

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37 

8742 

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22 

38 

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21 

38 

8875 

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894i 

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1241 

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18 

4i 

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42 

1  332 

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42 

94o4 

9426 

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17 

43 

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16 

43 

9536 

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16 

44 

1606 

1628 

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1720 

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44 

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1742 

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M 

45 

98oo 

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99io 

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46 

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46 

9932 

9954 

9976 

9998 

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47 

201  5 

2038 

2061 

2084 

2106 

2I29 

12 

47 

9.84oo64 

0086 

0108 

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0174 

12 

48 

2l52 

2I75 

2197 

222O 

2243 

2266 

I  I 

48 

oi96 

0218 

0240 

0262 

0284 

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II 

49 

2288 

23ll 

2334 

2356 

2379 

2402 

10 

49 

o328 

o35o 

0372 

o393 

o4i5 

o437 

10 

5o9.  832425 

2447 

2470 

2493 

25i5 

2538 

Q 

5o 

q.84o45q 

o48i 

o5o3 

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o547 

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9 

5i 

256i 

2584 

260612620 

2652 

2674 

8 

5i 

o59i 

o6i3 

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c678 

o7oo 

« 

52 

2697 

2720 

2742  2765 

2788 

2810 

7 

52 

0722 

0744 

0766 

o788 

0810 

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7 

53 

2833 

2856 

2878 

29OI 

2924 

2946 

6 

53 

o854 

0876 

o897 

0919 

o94i 

o963 

6 

54 

2969 

2992 

3oi4 

3o37 

3o6o 

3o82 

5 

54 

o985 

1007 

IO29 

io5i 

IO72 

io94 

5 

55 

3io5 

3i28 

3i5o 

3i73 

3i96 

32i8 

4 

55 

1116 

n38 

1160 

1182 

1204 

1226 

4 

56 

324i 

3263 

3286 

33o9 

333i 

3354 

3 

56 

1247 

1269 

I29I 

:3i3 

i335 

i357 

3 

57 

3377 

3399 

3422 

3444 

3467 

349o 

2 

57 

i378 

i4oo 

1422 

1  444 

i466  i488 

2 

58 

35i2 

3535 

3557 

358o 

36o3 

3625 

I 

58 

i5oq  i53i 

i553 

i575 

i5q7ii6iq 

I 

5, 

3648 

367o 

3693 

37i6 

3738 

376i 

O 

59 

i64o 

1662 

1  684 

1-706  i728|i749 

0 

60" 

50" 

40" 

30" 

20" 

10" 

B 

60" 

50"  |  40"  j  30"   20"  i  10" 

C3 

Co-sine  of  47 

Degrees. 

3 

Co-sine  of  46  Degrees. 

•9 

as 

•p      <  1"  2"  3''   4" 

5"  G"  7"  8"  9" 

,„,  (  1"  2"  3"  4"  5"  G"  7"  8"  9" 

artc  2  5  7  9 

11  14  1(5  18  21   A   ""I  2   4   7   9  11  13  16  18  20 

LOGARITHMIC    TANGENTS. 


67 


1 


Tangent  of  42  Degrees. 

— 
I 

Tangent  of  43  Degrees. 

« 

0" 

10"  |  20"   30"   40" 

50" 

* 

0"    j  10"   20" 

30"   40" 

50" 

0 

I 

9.954437 
469i 

448o 
4734 

4522 

4776 

4564 
4819 

4607 

486i 

4649 
49o3 

59 

58 

o 

i 

9.96965619698 
99099951 

974o 
9994 

9783 
..36 

9825 
..78 

9867 
.  120 

59 

58 

2 

4946 

4988 

5o3o 

5o73 

5n5 

5i57 

57 

2 

9.97oi62 

O205 

0247 

0289 

o33i 

o373 

57 

3 

5  200 

5a42 

5284 

5327 

5369 

5-4n 

56 

3 

o4i6 

o458 

o5oo 

0542 

o584 

0627 

56 

4 

5454|5496 

5538 

558i 

5623 

5665 

55 

4 

o669 

07II 

o753 

o796 

o838 

0880 

55 

5    57o8 

575o 

5792 

5835 

5877 

59i9 

54 

5 

0922 

o964 

IO07 

1049 

io9i 

n33 

54 

6 

596r 

6oo4 

6o46 

6088 

6i3i 

Oi73 

53 

6 

n75 

1218 

1260 

1302 

1  344 

i386 

53 

7 

62i5 

6258 

63oo 

6342 

6385 

b427 

52 

7 

i47i 

i5i3 

i555 

i597 

i64o 

52 

8 

6469 

65i2 

6554 

6596 

6639 

6681 

5i 

8 

1682 

i  -724 

i766 

1808 

i85i 

i893 

5i 

9 

6723 

6766 

6808 

685o 

6893 

6935 

5o 

9 

i935 

i977 

2OI9 

2062 

2IO4 

2146 

5o 

10 

9-9'6977 

702O 

7062  7104 

7i46 

7189 

49 

10 

9.972i88 

2230 

2273 

23i5 

2357 

2399 

49 

ii 

7231 

7273 

73i6 

7358 

7400 

7443 

48 

ii 

244  1 

2484 

2526 

2568 

2610 

2652 

48 

12 

7485 

7527 

757o 

7612 

7654 

7697 

47 

12 

2695 

2737 

2779 

2821 

2863 

29o5 

47 

i3 

7739 

7781 

7823 

7866 

79o8 

795o 

46 

i3 

2948 

2990 

3o32 

3o74 

3n6 

3i59 

46 

i4 

7993 

8o35 

8077 

8120 

8162 

8204 

45 

i4 

3201 

3243 

3285 

3327 

337o 

34i2 

45 

i5 

8247 

8289 

833! 

8373 

84i6 

8458 

44 

i5 

3454 

3496 

3538 

358i 

3623 

3665 

44 

16 

85oo 

8543 

8585 

8627 

8670 

87I2 

43 

16 

37o7 

3749 

379i 

3834 

3876 

39i8 

43 

17 

8754 

8796 

8839 

8881 

8923 

8966 

42 

17 

396o 

4O02 

4o45 

4o87 

4i29 

4i7i 

42 

18 

9oo8 

9o5o 

9o93 

9i35 

9i77 

9219 

4i 

18 

42  1  3 

4255 

4298 

434o 

4382 

4424 

4i 

19 

9262 

93o4 

9346 

9389 

943i 

9473 

4o 

19 

4466 

45o9 

455i 

4593 

4635 

4677 

4o 

20 

9.9595i6 

9558 

9600 

9642 

9685 

9727 

39 

20 

9.974720 

4762 

48o4 

4846 

4888 

493o 

39 

21 

9769 

98l2 

9854 

9896 

9938 

998i 

38 

21 

4973 

5oi5 

5o57 

5o99 

5i4i 

5i83 

38 

22 

9.96oo23 

oo65 

0108 

oi5o 

OI92 

0234 

37 

22 

5226 

5268 

53io 

5352 

5394 

5437 

3? 

23 

0277 

o3i9 

o36i 

o4o4 

o446 

o488 

36 

23 

5479 

552i 

5563 

56o5 

5647 

569o 

36 

24 

o53o 

o573 

o6i5 

o657 

o7oo 

0742 

35 

24 

5732 

5774 

58i6 

5858 

59oi 

5943 

35 

25 

0784 

0826 

o869 

o9n 

o953 

0996 

34 

25 

5985 

602-7 

6o69 

6m 

6i54 

6i96 

34 

26 

io38 

1080 

1122 

n65 

I2O7 

1249 

33 

26 

6238 

6280 

6322 

6364 

64o7 

6449 

33 

27 

I292 

i334 

i376 

i4i8 

i46i 

i5o3 

32 

27 

649i 

6533 

6575 

661-7 

6660 

6702 

32 

28 

1  545 

i587 

i63o 

1672 

i7i4 

i757 

3i 

28 

6744 

6786 

6828 

687o 

69i3 

6955 

3i 

29 

i799 

i84i 

i883 

I926 

i968 

2OIO 

3o 

29 

6997 

7o39 

-7081 

7I23 

-7166 

7208 

3o 

3o 

9.962O52 

2095 

2137 

2179 

2222 

2264 

29 

3o 

9.97725o 

7292 

7334 

7377 

74i9 

746  1 

29 

3i 

23o6 

2348 

239I 

2433 

2475 

25l7 

28 

3i 

75o3 

7545 

7587 

763o 

7672 

77i4 

28 

32 

256o 

2602 

2644 

2686 

2729 

2771 

27 

32 

7756 

7798 

784o 

7882 

7925 

7967 

27 

33 

28i3 

2856 

2898 

294o 

2982 

3o25 

26 

33 

8oo9 

8o5i 

8o93 

8i35 

8i78 

8220 

26 

34 

3o67 

3io9 

3i5i 

3i94 

3236 

3278 

25 

34 

8262 

83o4 

8346 

8388 

843i 

8473 

25 

35 

3520 

3363 

34o5 

3447 

3489 

3532 

24 

35 

85i5 

8557 

8599 

864i 

8684 

8726 

24 

36 

3574 

36i6 

3659 

3701 

3743 

3785 

23 

36 

8768 

8810 

8852 

8894 

8937 

8979 

23 

37 

3828 

387o 

39I2 

3954 

3997 

4o39 

22 

37 

9O2I 

9o63 

9io5 

9147 

9i9o 

9232 

22 

38 

4o8i 

4l23 

4i66 

4208 

4292 

21 

38 

9274 

93i6 

9358 

9400 

9442 

9485 

21 

39 

4335 

4377 

44i9 

446  1 

45o4 

4546 

20 

39 

9569 

96n 

9653 

9695 

9738 

20 

4o 

9.  964588 

463o 

4673 

47i5 

4757 

4799 

'9 

4o 

9.9-79-780 

9822 

9864 

99o6 

9948 

999° 

ig 

4i 

4842 

4884 

4926 

4968 

Son 

5o53 

18 

4i 

9.980033 

oo75 

OII7 

oi59 

0201 

0243 

18 

42 

SogS 

5i37 

5i8o 

5222 

5264 

53o6 

17 

42 

0286 

o328 

o37o 

0412 

0454 

o496 

17 

43 

5349 

539i 

5433 

54?5 

55i8 

556o 

16 

43 

o538 

o58i 

0623 

o665 

0707 

0749 

16 

44 

56o2 

5644 

5687 

5729 

577i 

58i3 

i5 

44 

°79* 

o834 

08-76 

o9i8 

0960 

IOO2 

i5 

45 

5855 

5898 

594o 

5982 

6024 

6067 

i4 

45 

1044 

1086 

II29 

n7i 

I2l3 

1255 

i4 

46 

6109 

6i5i 

6i93 

6236 

6278 

6320 

i3 

46 

I297 

i339 

1382 

1424 

i466 

i5o8 

i3 

47 

6362 

64o5 

6447 

6489 

653i 

6574 

12 

47 

i55o 

i592 

i634 

i677 

I7i9 

1761 

12 

48 

6616 

6658 

6700 

6742 

6785 

682-7 

II 

48 

i8o3 

i845 

i887 

I929 

1972 

2Ol4 

II 

49 

6869 

69n 

6954 

6996 

7o38 

7080 

IO 

49 

2o56 

2098 

2l4o 

2182 

2224 

2267 

IO 

5o 

9.96-7123 

7i65 

7207 

7249 

729I 

7334 

9 

5o 

9.982309 

235i 

2393 

2435 

2477 

25  I  9 

9 

5i 

7376 

74i8 

7460 

75o3 

7545 

7587 

8 

5i 

2562 

2604 

2646 

2688 

273o 

2772 

8 

52 

7629 

7672 

7714 

7756 

7798 

784o 

7 

52 

2814 

2857 

2899 

294i 

2983 

3o25 

17 

53 

7883 

7925 

7967 

8oo9 

8o52 

8o94 

6 

53 

3o67 

3io9 

3i52 

3i94 

3236 

3278 

6 

54 

8i36 

8178 

8220 

8263 

83o5 

8347 

5 

54 

3320 

3362 

34o4 

3447 

3489 

353i 

5 

55 

8389 

8432 

8474 

85i6 

8558 

8600 

4 

55 

3573 

36i5 

3657 

3699 

374z 

3784 

4 

56 

8643 

8685 

8727 

8769 

8812 

8854 

3 

56 

3826 

3868 

39io 

3952 

3994 

4o37 

3 

57!    8896 

8938 

898o 

9023 

9o65 

9107 

2 

57 

4o79 

4l2I 

4i63 

4i$o5 

4247 

4289 

2 

58 

9149 

9I92 

9234 

9276 

93i8 

9360 

I 

58 

4332 

4374 

44i6 

4458 

45oo 

4542 

I 

59 

94o3 

9445 

9487 

9529 

957i 

9614 

0 

59 

4584|4627 

4669 

4711 

4753 

4795 

o 

60" 

50" 

40" 

30" 

20" 

10" 

a 

60"     50"   40"   30" 

20" 

10" 

Co-tangent  of  47  Degrees. 

.9 

Co-tangent  of  46  Degrees.    § 

C  1"  2"  3"  4"  5"  6"  7"  8"  9" 

t  S  i"  2"  3"  4"  5"  6"  7"  8"  9" 

P.  Part  J  4   8  13  17  oj  05  3j  34  38 

1  .Jart£  4   8  13  J7  21  25  30  34  3g 

68 


LOGARITHMIC 


I  \  E  o. 


17 

Sine  of  44  Degrees. 

d 

Sine  of  45  Decrees. 

r 

3 

0' 

10" 

20"  [  30"  •  40" 

50" 

€ 

0" 

10" 

20" 

30"  ]  40'' 

50" 

b 

9-841771 

i793 

i8i5;i837ii858 

1880 

59 

o 

9.849485 

95o6 

9527 

9548  9569 

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5 

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4 

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56 

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3 

56 

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6466  6486  6507 

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3 

57 

9106 

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9i48 

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2 

57 

6568  6588  66o9  6629  6649  6670 

2 

58 

9232 

9253 

92749295 

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58 

6690  6710  6731  67?  i  6771  6792 

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59 

9359 

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94oi 

9422 

9443:9464 

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59 

6812  6832  6853!6873  6893  69i4 

0 

eo" 

50" 

40"   30" 

20"  I  10" 

C 

60"    |  50"   40"   30"   20"   10" 

j 

Co-sine  of  45  Degrees. 

3 

Co-sine  of  44  Degrees.     % 

p  p  .<  1"  2"  3"  4"  5"  6"  7"  8"  9" 

.  (  1"  2"  3"  4"  5"  6"  7"  8"  9" 

irt$  2   4   G   9  11  13  15  17  19 

irt(  2  4   G   8  10  12  14  17  19 

LOGARITHMIC    TANGENTS. 


60 


"7 

Tangent  of  44  Degrees. 

.5 

Tangent  of  45  Degrees. 

•* 

0" 

10" 

20"  |  30" 

40"  |  50" 

m 

0"      10"  |  20" 

30" 

40"  |  50" 

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9.984837 

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7726 

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b 

5a 
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6 

52 

53 

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3393 

3i83 
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6 

54 

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8568 

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5 

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3688 

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3772 

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3857 

5 

55 

873718779 

8821 

8863 

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4 

55 

3899 

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3983 

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4o67 

4io9 

4 

56 

8989 

9o3i 

9074 

9116 

9i58  9200 

3 

56 

4i52 

4ig4 

4236 

4278 

4320 

4362 

3 

$7 

9242 

9284 

9326 

9368 

94io  9453 

2 

57 

44o4 

4447 

4489 

453i 

4573 

46i5 

a 

58 

9495 

9537 

9579 

9621 

9663  97o5 

I 

58 

4657 

4699 

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4784 

4826 

4868 

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59 

9747 

9789 

9832 

9874 

9916)9958 

O 

59 

49io 

4952 

4994 

5o36 

5o79  5i2i 

o 

60" 

50"   40" 

30" 

20"   10" 

r 

60" 

50" 

40" 

30" 

20"  |  10" 

d 

Co-tangent  of  45  Degrees. 

& 

Co-tangent  of  44  Degrees. 

1 

P  Part*  r/  2/<  3//  4//  5"  6"  7"  8"  9" 

irll  4   8  13  17  21  25  29  34  38 

,<  1"  2"  3"  4"  5"  6"  7"  8"  9' 
.  |  4   8  13  17  21  25  29  34  38 

LOGARITHMIC 


a  J      Sino  of  46  Degrees. 

T  
d 

Sine  of  47  Degrees. 

M  1    o" 

10"   20"   30" 

40"  |  50"  j 

2 

0" 

10" 

20"  |  30" 

40" 

50" 

0)9.  85693^ 

6954 

6975 

6995 

7oi5 

7o36  59 

o9.864i27 

4i47 

4i67 

4i86 

4206 

4226 

59 

i 

7o56 

7076 

7°97 

7117 

7i37 

7i58 

58 

i 

4245 

4265 

4284 

43o4 

432< 

4343 

58 

2 

7178 

7I98 

72i9 

7239 

7259 

7279 

57 

2 

4363 

4383 

4402 

4422 

444  1 

446i 

57 

3 

73oo 

7320 

734o 

73  6  1 

738i 

74oi 

56 

2 

448  1 

45oo 

4520 

4539 

4559 

4579 

56 

4 

7422 

7442 

7462 

7^82 

75o3 

7523 

55 

t 

4598 

46i8 

4637 

4657 

4676 

4696 

55 

5 

7543 

7563 

7584 

7po^ 

7624 

7645 

54 

r 

4716 

4735 

4755 

4774 

4794 

48i3 

54 

5 

7665 

7685 

77o5 

7p6 

7746 

7766 

53 

6 

4833 

4853 

4872 

4892 

49n 

493i 

53 

7 

7786 

7807 

7827 

7&47 

7867 

7888 

52 

7 

495o 

497o 

4990 

5oo9 

5o29 

5o48 

52 

8 

7908 

7928 

7948 

7968 

7989 

8009 

5i 

8 

5o68 

5o87 

5io7 

5i26 

5i46 

5i65 

5i 

9 

8029 

8o49 

8o7o|8o9o 

8110 

8i3o 

5o 

9 

5i85 

52o4 

5224 

5244 

5263 

5283 

5o 

10 

9.858i5i 

8171 

8191 

8211 

823i 

8252 

49 

10 

9.  8653  02 

5322 

5341 

536i 

538o 

54oo 

49 

n 

8272 

8292 

83128332 

8353 

8373 

48 

ii 

54i9 

5439 

5458 

5478 

5497 

55i7 

48 

12 

/  SSgS 

84i3 

84^3  8454 

8474 

8494 

47 

12 

5536 

5556 

5575 

5595 

56i4 

5634 

47 

i3 

85i4 

8534 

8554  8575 

8595 

86i5 

46 

i3 

5653 

5673 

5692 

57I2 

573i 

575i 

46 

i4 

8635 

8655 

8675  8696 

87i6 

8736 

45 

i4 

577o 

579o 

58  o9 

5828 

5848 

5867 

45 

i5 

8756 

8776 

879688i7 

8837 

8857 

44 

i5 

5887 

59o6 

5926 

5945 

5965 

5984 

44 

16 

8877 

8897 

89i78937 

8958 

8978 

43 

16 

6oo4 

6023 

6042 

6062 

6081 

6101 

43 

r? 

8998 

9018 

9o38  9o58 

9o78 

9098 

42 

ij 

6120 

6i4o 

6i59 

6i79 

6i98 

62I7 

42 

18- 

9119 

9i39 

9i59'9i79 

9i99 

9219 

4i 

18 

6237 

6256 

6276 

6295 

63i5 

6334 

4i 

r9 

9239 

9259 

9279  93oo 

9320 

934o 

4o 

I9 

6353 

6373 

6392 

6412 

643  1 

645o 

4o 

209.859360 

9380 

94oo  942o 

944o 

946o 

39 

20 

9.866-47o 

6489 

65o9 

6528 

6547 

6567 

39 

21 

9480 

95oi 

952i  954i 

956i 

958i 

38 

21 

6586 

6606 

6625 

6644 

6664 

6683 

38 

22 

9601 

9621 

964i  9661 

9681 

97oi 

37 

22 

67o3 

6722 

674i 

676i 

678o 

6800 

37 

23 

9721 

974i 

976il978i 

9802 

9822 

36 

23 

6819 

6838 

6858 

6877 

6896 

69i6 

36 

-4 

9842 

9862 

9882  9902 

9922 

9942 

35 

24 

6935 

6954 

6974 

6993 

7oi3 

7o32 

35 

25 

9962 

9982 

.  .  .2  .  .22 

..42 

..62 

34 

25 

7o5i 

7071 

7o9o 

7io9 

7I29 

7i48 

34 

26 

9.860082 

OIO2 

OI22'oi42 

0162 

0182 

33 

26 

7167 

7187 

72o6 

7225 

7245 

7264 

33 

27 

O2O2 

O222 

O242  O262 

0282 

0302 

32 

27 

7283 

73o3 

7322 

734i 

736i 

738o 

32 

28 

O322 

0342 

03620382 

04O2 

0422 

3i 

28 

7399 

74i9 

7438 

7457 

7476 

7496 

3i 

29 

0442 

O462 

0482,  o5o2 

O522 

0542 

3o 

29 

75i5 

7534 

7554 

7573 

7592 

76l2 

3o 

3o 

9.86o562 

o582 

0602  0622 

0642 

0662 

29 

3o 

9.86763i 

765o 

7669 

7689 

77o8 

7727 

29 

3i 

0682 

0702 

O722  O742 

0762 

0782 

28 

3i 

7747 

7766 

7785 

78o4 

7824 

7843 

28 

32 

0802 

0822 

o842!o862 

0882 

0902 

27 

32 

•7862 

7882 

79oi 

7920 

7939 

7959 

27 

33 

0922 

0941 

o96i  o98i 

IOOI 

1021 

26 

33 

7978 

7997 

8016 

8o36 

8o55 

8o74 

26 

34 

io4i 

1061 

1081  noi 

II2I 

Il4l 

25 

34 

8o93 

8n3 

8i32 

8i5i 

8i7o 

8i9o 

25 

35 

1161 

1181 

I2OI  1221 

1240 

I26O 

24 

35 

8209 

8228 

8247 

8267 

8286 

83o5 

24 

36 

1280 

i3oo 

i32o  i34o 

i36o 

i38o 

23 

36 

8324 

8343 

8363 

8382 

84oi 

8420 

23 

37 

i4oo 

l42O 

i439  i459 

i479 

1499 

22 

37 

844o 

8459 

8478 

8497 

85!6 

8536 

22 

38 

i5i9 

i539 

i559  i579 

i599 

1618 

21 

38 

8555 

8574 

8593 

8612 

8632 

865i 

21 

39 

i638 

i658 

i678:i698 

I7i8 

i738 

2O 

39 

867o 

8689 

87o8 

8728 

8747 

8766 

2O 

4o 

9.  86:758 

1777 

i797  1817 

i837 

i857 

I9 

4o 

9.868785 

8804 

8823 

8843 

8862 

8881 

J9 

4i 

1877 

i897 

I9i6  I936 

i956  i976 

18 

4i 

8900 

89i9 

8939 

8958 

8977 

8996 

18 

42 

i996 

2016 

2o35  2o55 

2075 

2095 

J7 

42 

9015 

9o34 

9o53 

9°73 

9092 

9in 

17 

43 

2Il5 

2i35 

2  1  54  21  74 

2I94 

22l4 

16 

43 

9i3o 

9i49 

9i68 

9i88 

9207 

9226 

16 

44 

2234 

2254 

2273J2293 

23i3 

2333 

i5 

44 

9245 

9264 

9283 

9302 

932I 

934i 

i5 

45 

2353 

2372 

2392 

2412 

24312 

2452 

i4 

45 

g36o 

9379 

9398 

94i7 

9436 

9455 

i4 

46 

2471 

249I 

25ll 

253i 

255i 

2570 

i3 

46 

9474 

9494 

95i3 

9532 

955i 

957o 

i3 

47 

259o 

26lO 

263o 

265o 

2669 

2689 

12 

47 

9589 

96o8 

9627 

9646 

9665 

9685 

12 

48 

2709 

2729 

2748 

2768 

2788  2808 

II 

48 

97o4 

9723 

9742 

976i 

978o 

9799 

II 

49 

2827 

2847 

2867 

2887 

2906  2926 

IO 

49 

9818 

9837 

9856 

9875 

9894 

99i4 

IO 

5o 

9.862946 

2966 

2985 

3oo5 

3o25l3o45 

9 

5o 

9.869933 

9952 

9971 

999° 

...9 

..28 

9 

5i 

3o64 

3o84 

3io4 

3i24 

3i43 

3i63 

8 

5i 

9.87oo47 

0066 

oo85 

oio4 

0123 

Ol42 

8 

52 

3i83 

32o3 

3222 

3242 

3262 

3281 

7 

52 

0161 

0180 

oi99 

0218 

0238 

O257 

7 

53 

33oi 

332i 

334i 

336o 

338o 

34oo 

6 

53 

O276 

0295 

o3i4 

o333 

o352 

o37i 

6 

54 

34i9 

3439 

3459 

3478 

3498 

35i8 

5 

54 

o39o 

0409 

0428 

o447 

o466 

o485 

5 

55 

3538 

3557 

3577 

3597 

36i6 

3636 

4 

55 

o5o4 

o523 

o542 

o56i 

o58o 

o599 

4 

.r>6 

3656 

3675 

3695 

37i5 

3734 

3754 

3 

56 

0618 

o637 

o656 

o675  o694 

o7i3 

3 

57 

3774 

3793 

38i3 

3833 

3852 

3872 

2 

57 

0782 

o75i 

o77o 

o789  0808 

o827 

2 

58 

3892 

39n 

393i 

395i 

397o 

399o 

I 

58 

o846  o865  o884  o9o3  o922 

o94i 

I 

59 

4oio 

4029 

4o49 

4o69 

4o88 

4io8 

0 

59 

o96o  o979  o998  ioi7  io36  io54 

O 

60" 

50" 

40" 

30"  I  20" 

10" 

c 

60"     50"   40"   30"   20"   10"  |  e- 

Co-sine  of  43  Degrees. 

2 

Co-sine  of  42  Degrees. 

.Cl"  2"  3"  4"  5"  6"  7"  8"  9" 

p  p  tf  1"  2"  3"  4"  5"  6"  7"  8"  0" 

lrt}  2   4   G   8   10  12  14  16  18 

1  I  2   4   6   8  10  12  14  15  17 

LOGARITHMIC    TANGENTS. 


a  '  Tangent  of  46  Degrees. 

d 

Tangent  of  47  Degrees. 

•S  J     0"   '  |  10" 

20"~J~30" 

40" 

50" 

2 

0" 

10"   ?0" 

30" 

40" 

50" 

;j 

1  o 

io.oi5i63 

52o5 

5247 

5289 

533i 

5373 

59 

0 

io.o3o344 

o386 

0429 

0471 

o5i3 

o555 

59 

I 

54i6 

5458 

55oo 

5542 

5584 

5626 

58 

I 

0597 

o64o 

0682 

0724 

0766 

0808 

58 

2 

5668 

5711 

5753 

5795 

5837 

5879 

57 

2 

o85i 

o893 

o935 

0977 

1020 

1062 

57 

3 

5921 

5963 

6006 

6o48 

6090 

6i32 

56 

3 

1  104 

n46 

1188 

I23l 

1273 

i3i5 

56 

4 

6i74 

6216 

6258 

63oi 

6343 

6385 

55 

4 

i357 

i4oo 

i442 

i484 

i526 

1  568 

55 

5 

6427 

6469 

65n 

6553 

6596 

6638 

54 

5 

1611 

i653 

1695 

i737 

1780 

1822 

54 

6 

6680 

6722 

6764 

6806 

6848 

689i 

53 

6 

1864 

1906 

i948 

1991 

2033 

2075 

53 

7 

6933 

6975 

701-7 

7o59 

7101 

7i43 

52 

7 

2117 

2160 

2202 

2244 

2286 

2328 

5a 

8 

7186 

7228 

7270 

73l2 

7354 

7396 

5i 

8 

2371 

24i3 

2455 

2497 

254o 

2582 

5i 

9 

7438 

748  1 

7523 

7565 

7607 

7649 

5o 

9 

2624 

2666 

2709 

275l 

2793 

2835 

5o 

10 

10.017691 

7733 

7776 

78i8 

7860 

79Q2 

49 

10 

10.032877 

2920 

2962 

3oo4 

3o46 

3o89 

49 

ii 

7944 

7986 

8028 

8o7i 

8ii3 

8i55 

48 

ii 

3i3i 

3i73 

32i5 

3258 

33oo 

3342 

48 

i  12 

8197 

8239 

8281 

8323 

8366 

84o8 

47 

12 

3384 

3426 

3469 

35n 

3553 

3595 

47 

i3 

845o 

8492 

8534 

8576 

8618 

8661 

46 

i3 

3638 

368o 

3722 

3764 

38o7 

3849 

46 

i4 

8703 

8745 

8787 

8829 

8871 

8914 

45 

i4 

3891 

3933 

3976 

4oi8 

4o6o 

4l02 

45 

i5 

8956 

8998 

9o4o 

9o82 

9124 

9166 

44 

i5 

4i45 

4i87 

4229 

427I 

43i3 

4356 

44 

16 

9209 

925i 

9293 

q335 

9377 

9419 

43 

16 

4398 

444o 

4482 

4525 

4567 

4609 

43 

»7 

9462 

95o4 

9546 

o588 

963o 

9672 

42 

I? 

465  1 

4694 

4?36 

4778 

4820 

4863 

42 

18 

9714 

9757 

9799 

984i 

9883 

9925 

4i 

18 

4905 

4947 

4989 

5o32 

5o74 

5n6 

4i 

J9 

9967 

.  .10 

..62 

..94 

.136 

.178 

4o 

J9 

5i58 

52OI 

5243 

5285 

5327 

537o 

4o 

20 

IO.O20220 

0262 

o3o5 

0347 

o389 

o43i 

39 

20 

io.o354i2 

5454 

5496 

5539 

558i 

5623 

39 

21 

0473 

o5i5 

o558 

0600 

0642 

o684 

38 

21 

5665 

5708 

5750 

5792 

5834 

5877 

38 

22 

0726 

0768 

0810 

o853 

o895 

o937 

37 

22 

6919 

5961 

6oo3 

6o46 

6088 

6i3o 

37 

23 

0979 

1021 

io63 

1106 

n48 

II9O 

36 

23 

6172 

62i5 

6257 

6299 

634i 

6384 

36 

24 

1232 

1274 

i3i6 

i359 

i4oi 

i443 

35 

24 

6426 

6468 

65n 

6553 

6595 

6637 

35 

25 

i485 

1527 

i569 

1612 

i654 

i696 

34 

25 

6680 

6^22 

6764 

6806 

6849 

6891 

34 

26 

1738 

1780 

1822 

i865 

1907 

i949 

33 

26 

6933 

6975 

7018 

7o6o 

7I02 

7i44 

33 

27 

1991 

2033 

2075 

2118 

2160 

22O2 

32 

27 

7187 

7229 

7271 

73i4 

7356 

7398 

32 

28 

2244 

2286 

2328 

2370 

24l3 

2455 

3i 

28 

744o 

7483 

7525 

7567 

76o9 

7652 

3i 

29 

2497 

2539 

258i 

2623 

2666 

27o8 

3o 

29 

7694 

7736 

7778 

782I 

7863 

79o5 

3o 

3c 

10.022750 

2792 

2834 

2877 

2919 

296i 

29 

3o 

10.037948 

799° 

8o32 

8o74 

8117 

8i59 

29 

3i 

3oo3 

3o45 

3o87 

3i3o 

3172 

3214 

28 

3i 

8201 

8243 

8286 

8328 

837o 

84i3 

28 

32 

3256 

3298 

334o 

3383 

3425 

3467 

27 

32 

8455 

8497 

8539 

8582 

8624 

8666 

27 

33 

35og 

355i 

3593 

3636 

3678 

3720 

26 

33 

8708 

875i 

8793 

8835 

8878 

8920 

26 

34 

3762 

38o4 

3846 

3889 

3931 

3973 

25 

34 

8962 

9004 

9047 

9089 

9i3i 

9174 

25 

35 

4or5 

4o57 

4o99 

4i42 

4i84 

4226 

24 

35 

9216 

9258 

gSoo 

9343 

9385 

9427 

24 

36 

4268 

43io 

4353 

4395 

4437 

4479 

23 

36 

9470 

9512 

9554 

9596 

9639 

9681 

23 

3? 

452i 

4563 

46o6 

4648 

4690 

4732 

22 

37 

9723 

9766 

9808 

985o 

9892 

9935 

22 

38 

4774 

48i7 

4859 

4901 

4943 

4985 

21 

38 

9977 

..19 

..62 

.  io4 

.i46 

.188 

21 

39 

5027 

6070 

5lI2 

5i54 

5i96 

5238 

20 

39 

io.o4o23i 

0273 

o3i5 

o358 

o4oo 

0442 

2O 

4o 

10.025280 

5323 

5365 

54o7 

5449 

549i 

19 

4o 

io.o4o484 

0527 

0569 

061  1 

o654 

0696 

19 

4i 

5534 

5576 

56i8 

566o 

5702 

5745 

18 

4i 

o738 

0781 

0823 

o865 

0907 

0960 

18 

42 

5787 

5829 

5871 

59i3 

5955 

5998 

J7 

42 

O992 

io34 

1077 

ni9 

1  161 

1204 

*7 

43 

6o4o 

6082 

6124 

6166 

6209 

625i 

16 

43 

1246 

1288 

i33o 

i373 

i4i5 

1457 

16 

44 

6293  6335 

6377 

64i9 

6462 

65o4 

i5 

44 

i5oo 

i542 

i584 

1627 

1669 

I7II 

i5 

45 

6546 

6588 

663o 

6673 

67i5 

6757 

i4 

45 

I753 

1796 

i833 

1880 

1923 

i965 

i4 

46 

6799 

684i 

6884 

6926 

6968 

7OIO 

i3 

46 

2007 

2o5o 

2092 

2134 

2177 

2219 

i3 

4? 

7052 

7o95 

7i37 

7i79 

7221 

7263 

12 

4? 

226l 

23o3 

2346 

2388 

243o 

2473 

12 

48 

73o5 

7348 

739o 

7432 

7474 

75i6 

I  I 

48 

25i5 

2557 

2600 

2642 

2684 

2727 

II 

49 

7559 

7601 

7643 

7685 

7727 

7770 

IO 

49 

2769 

2811 

2854 

280.6 

2938 

2980 

IO 

5o 

10.027812 

7854 

7896 

7938 

798i 

8o23 

9 

5o 

io.o43o23 

3o65 

3107 

3i5o 

3192 

3234 

9 

5i 

8o65 

8107 

8i49 

8l92 

8234 

8276 

8 

5i 

3277 

33I9 

336i 

34o4 

3446 

3488 

8 

52 

83i8 

836o 

84o3 

8445 

8487 

8529 

7 

52 

353i 

3573 

36i5 

3658 

3700 

3742 

7 

53 

857i 

86i4 

8656 

8698 

874o 

8782 

6 

53 

3785 

3827 

3869 

39I2 

3954 

3996 

6 

54 

8825 

8867 

8909 

895i 

8993 

9o36 

5 

54 

4o39 

4o8i 

4123 

4i65 

4208 

425o 

5 

55 

9078 

9I20 

9162 

9204 

9247 

9289 

4 

55 

4292 

4335 

4377 

44i9 

4462 

45o4 

4 

55 

933i 

9373 

94i6 

9458 

95oo 

9542 

3 

56 

4546 

458o 

463i 

4673 

4716 

4758 

3 

57 

9584 

9627 

9669 

9711 

9753 

9795 

2 

57 

48oo 

4843 

4885 

4927 

4970 

5012 

2 

Cft 

9838 

9880 

99220064 

...6 

..49 

I 

58 

5o54 

5o97 

SiSg 

5i8i 

5224 

5a66 

I 

59 

io.o3oo9i 

oi33 

oi75 

0217 

0260 

0302 

O 

59 

53o9 

535i 

53g3 

5436 

5478 

5520 

O 

60" 

50" 

40" 

30"   20" 

10" 

H 

60"     50" 

40" 

30" 

ao" 

10" 

^ 

Co-tangent  of  43  Decrees. 

s 

Co-tangent  of  42  Degrees. 

.£? 

E 

p  „   (  1"  2"  3"  4"  .V  6"  7"  8"  9" 

P  Part  J  l"  ~"  3"  4"  5"  6"  7"  8//  'J" 

/  4   8  13  17  21  25  30  34  38 

>  4   8  1?  r  ','1  25  30  34  38 

j 

LOGARITHMIC    IS  i  N  E  s. 


J3 

Sine  of  48  Degrees. 

A 

Sine  of  49  Degrees. 

2 

0"    |  10' 

20"   30"   40' 

50' 

& 

0' 

10"   20"   30" 

40' 

50" 

O 

9.87io73|io92 

ii  1  1  i  i3o  1149 

1168 

59 

o9.87778o 

779878i6 

7835 

7853 

7871 

5o 

I 

1187 

1206 

1225  1244  1263 

1282 

58 

i 

789o 

79o8 

7926 

7945 

7963 

7981 

58 

2 

i3oi 

1320 

1339  i358  i377 

i39557 

2 

7999 

8018 

8o36 

8o54 

8o72 

8o9i 

57 

3 

i4i4 

i433 

i452  i47i;i49o 

i5o9  56 

5 

8io9 

8127 

8i46 

81648182 

8200 

56 

4|    i528 

1547 

i566  i585  1604 

1622 

55 

L 

82I9 

8237 

8255 

8273 

8292 

83io 

55 

D 

i64i 

1660 

1679  1698  1717 

i736 

54 

t 

8328 

8346 

8365 

8383 

84oi 

84i9 

54 

£ 

i755 

1774 

1793  1811  i83o 

i849 

53 

6 

8438 

8456 

8474 

8492 

85n 

8529 

53 

7 

1868 

1887 

1906  1925  1944 

I962 

52 

7 

8547 

8565 

8583 

8602 

8620 

8638 

52 

8 

1981 

2000  2019  2o38  2057 

2076 

5i 

8 

8656 

8675 

8693 

8711 

8729 

8747 

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9 

2095 

2Il3 

2l32  2l5l|2I70 

2l89 

5o 

9 

8766 

8784 

8802 

8820 

8838 

8857 

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1019.872208 

2226 

2245  2264  2283 

2302 

49 

10 

9.878875 

8893 

8911 

8929 

8948 

8966 

49 

1  1 

2^21 

2340 

235823772396 

24i5 

48 

ii 

8984 

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9020 

9o39 

9o57 

9o75 

48 

12 

2434 

2452 

2471 

2490,2509 

2528 

47 

12 

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9III 

9129 

9i48 

9i66 

47 

i3 

2547 

2565; 

25342603:2622 

2641 

46 

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922O 

9238 

9257 

9275 

9293 

46 

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2659 

2678269727162735 

2753 

45 

i4 

93n 

9329 

9347 

9365 

93S4 

9402 

45 

i5 

2772 

2791! 

2810  2829 

2847 

2866 

44 

i5 

9420 

9438 

9456 

9474 

9492 

95n 

44 

16 

2885 

2904  2923  2941 

296o 

2979 

43 

16 

9529 

9547 

9565 

9583 

96oi 

9619 

43 

17 

2998 

3oi6  3o35  3o54 

3o73 

3091 

42 

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9637 

9656 

9674 

9692 

97io 

9728 

42 

18 

3no 

3i293i483i66 

3i85 

3204 

4i 

18 

9746 

9764 

9782 

9800 

9837 

4i 

19 

3223 

324l 

32603279 

3298 

33i6 

4o 

19 

9855 

9873 

989i 

9909 

9927 

9945 

4o 

20 

9.  873335 

3354 

3373  3391 

34io 

3429 

39 

20 

9.879963 

998! 

..18 

..36 

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39 

21 

3448 

3466 

3485  35o4 

3522 

354i 

38 

21 

9.88oo72 

0090 

0108 

0126 

oi44 

0162 

38 

22 

356o 

3579 

3597  36i6 

3635 

3653 

37 

22 

0180 

0198 

0216 

0234  0253 

0271 

37 

23 

3672 

369i 

37io:3728 

3747 

3766 

36 

23 

0289 

o3o7 

o325 

o343  o36i 

o379 

36 

24 

3784 

38o3 

3822384o 

3859 

3878 

35 

24 

o397 

o4i5 

o433 

o45i 

0469 

0487 

35 

25 

3896 

39i5 

393453953 

397i 

399o 

34 

25 

o5o5 

o523 

o54i 

o559 

o577 

o595 

34 

26 

4oo9 

4027  4o46j4o65 

4o83 

4lO2 

33 

26 

o6i3 

o63i 

o649 

0667 

0686 

0704  33 

27 

4l2I 

4i394i58 

4177 

4i95 

4214 

32 

27 

O722 

o74o 

o758 

0776 

0794 

0812  32 

28 

4232 

425i  4270 

4288 

43o7 

4326 

3i 

28 

o83o 

o848 

0866 

o8S4 

O902 

0920 

3i 

29 

4344 

43634382 

44oo 

44i9 

4438 

3o 

29 

0938 

0956 

o974 

0992 

IOIO 

1028 

3o 

3o 

9.874456 

4475  4493 

45i2 

453i 

4549 

29 

3o 

9.881046 

io63 

1081 

1099 

1117 

n35 

29 

3i 

4568 

4586  46o5 

4624 

4642 

466i 

28 

3i 

:i53 

ii7i 

n89 

1207 

1225 

1243 

28 

32 

468o 

46984717 

4735 

4754 

4773 

27 

32 

1261 

I279 

I297 

i3i5 

i333 

i35i 

27 

33 

4791 

48104828 

4847 

4866 

4884 

26 

33 

i369 

i387 

i4o5 

1423 

i44i 

i459 

26 

34 

4903 

4921 

4g4o 

4958 

4977 

4996 

25 

34 

1477 

i495 

l5l2 

i53o 

1  548 

i566 

25 

35 

5oi4 

5o335o5i 

5070 

5o88 

5io7 

24 

35 

1  584 

1602 

1620 

i638 

i656 

1674 

24 

36 

5i26 

5i445i63 

5i8i 

5200 

52i8 

23 

36 

1692 

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1728 

1746 

i763 

1781 

23 

37 

5237 

5255  5274 

5293 

53ii 

533o 

22 

37 

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1817 

i835 

i853 

1871 

1889 

22 

38 

5348 

53675385 

54o4 

5422 

544  1 

21 

38 

1907 

I925 

I942 

1960 

i978 

1996 

21 

39 

5459 

5478  5496 

55i5 

5534 

5552 

2O 

39 

20l4 

2032 

2o5o 

2068 

2086 

2103 

2O 

4o 

9.875571 

5589 

56o8 

5626 

5645 

5663 

19 

4o 

9.882I2I 

2l39 

2l57 

2I?5 

2I93 

2211 

19 

4i 

5682 

5700  5719 

5737 

57565774 

18 

4i 

2229 

2246 

2264 

2282 

23oo 

23i8 

18 

42 

5793 

58ii  583o 

5848 

58675885 

'7 

42 

2336 

2354 

237I 

2389 

2407 

2425 

17 

43 

5904 

5922  5g4i 

5959 

59786996 

16 

43 

2443 

2461 

2479 

2496 

25i4 

2532 

16 

44 

6oi4 

6o33  6o5i 

6o7o 

60886107 

i5 

44 

255o 

2568 

2586 

26o3 

2621 

2639 

i5 

45 

6i25 

6i44  6162 

6181 

6-1  99  6218 

i4 

45 

2657 

2675 

2692 

2710 

2728 

2746 

i4 

46 

6236 

6255  6273 

629I 

63io6328 

i3 

46 

2764 

2782 

2799 

2817 

2835 

2853 

i3 

47 

6347 

6365  6384 

6402 

6421  6439 

12 

47 

287I 

2888 

29o6 

2924 

2942 

2960 

12 

48 

6457 

6476  6494 

65i3 

653i 

655o 

II 

48 

2977 

2995 

3oi3 

3o3i 

3o49 

3o66 

II 

49 

6568 

6586  66o5 

6623 

6642  6660 

IO 

49 

3o84 

3  1  02 

3l2O 

3i37 

3i55 

3i73 

IO 

5o 

9.876678 

6697  6715 

6734 

6762  6770 

9 

5o 

9.883191 

32O9 

3226 

3244 

3262 

328o 

9 

5i 

6789 

6807  6826 

6844 

68626881 

8 

5i 

3297 

33i5 

3*333 

335i 

3368 

3380 

8 

52 

6899 

69186936 

6954 

6973  699i 

7 

52 

34o4 

3422 

3439 

3457 

3475 

3493 

7 

53 

7010 

7028  7046 

7065 

7o83j7ioi 

6 

53 

35io 

3528 

3546 

3564 

358i 

3599 

6 

54 

7120 

7i38  7i57 

7175 

7193  7212 

5 

54 

36i7 

3635 

3652 

367o 

3688 

37o5 

5 

55 

7230 

7248  7267 

7285 

73o3  7322 

4 

55 

372?  374i 

3759 

3776 

3794 

38i2 

4 

56 

734o 

73587877 

7?,95 

74i37432 

3 

56 

3829 

3847 

3865 

3883 

39oo 

39i8 

3 

57 

745o 

7468  7487 

75o5 

7523  754s 

2 

57 

3936 

3953 

397i 

3989 

4oo6 

4024 

2 

58 

756o 

7578:759776i5 

7633;7652 

I 

58 

4o42 

4o6o 

4077 

4095  4n3 

4i3o 

I 

59 

7670 

7688  77o7i7725 

77437762 

O 

59 

4i48 

4i664i83 

4201  4219  4236 

O 

Uw 

50"  | 

40"   30" 

20"   10" 

rt 

63" 

50"   40"   30"   20"   10" 

. 

Co-sine  of  4  1  Degrees. 

.8 

Co-sine  of  40  Degrees.     & 

(  i//  o/.  3" 

4"  5"  6"  7"  8"  9" 

P  PartJ1"  2"  3"  4"  5"  6"  7"  &>  °" 

^  <  2   4   6 

7   9  11  13  15  17 

1  \  24   5   7   9  11  13  14  1»> 

_L  O  r,  A  R  I  T  H  M  I  C      T  A  N  G  E  A  T  6. 


i 

Tangent  of  48  Decrees. 

a 

§ 

Tangent  of  49  Degrees. 

V    \  10"   20"   30"  j  40"   50" 

0"     i(y- 

20" 

30" 

40" 

50" 

0 

io.o45563 

56o5 

5647 

5690 

5732 

5774 

59 

o 

10.060837 

0879 

0922 

0966 

1007 

io5o 

59 

I 

6817 

5859 

5901 

5944 

5986 

6028 

58 

I 

1092 

n35 

II77 

I22O 

1262 

i3o5 

58 

2 

3 

6071 
6325 

6n3 
6367 

6i55  6198 
6409  6452 

6240 
6494 

6282 
6537 

57 
56 

r. 

i347 
1602 

1390 
i645 

1432 

1688 

i475 
1730 

i5I? 
I773 

i56o 
1.815 

57 
56 

41    6679 

6621 

6664 

6706 

6748 

6791 

55 

L 

i858 

1900 

1943 

1986 

2028 

2070 

55 

5     6833 

6875 

6918 

6960 

7002 

7o45 

54 

r. 

2Il3 

2i55 

2198 

224l 

2283 

2326 

54 

6 

7087 

7i3o 

7172 

7214 

7267 

7299 

53 

6 

2368 

2411 

2453 

2496 

2538 

258i 

53 

7 

734i 

7384 

7426 

7468 

7611 

7553 

52 

7 

2623 

2666 

2709 

275i 

2794 

2836 

62 

8 

7595 

7638 

7680 

7723 

7765 

7807 

5i 

8 

2879 

2921 

2964 

3oo6 

3o49 

3o92 

5i 

9 

7800 

7892 

7934 

7977 

8019 

8062 

5o 

9 

3  1  34 

3  1  77 

3219 

3262 

33o4 

3347 

5o 

10 

io.o48io4 

81468189 

823i 

8273 

83  1  6 

49 

10 

10.063389 

3432 

3475 

35i7 

356o 

36o2 

49 

ii 

8358 

84co!8443 

8485 

8528 

8070 

48 

ii 

3645 

3687 

373o 

3773 

38i5 

3858 

48 

12 

i3 

8612 
8867 

8655 
8909 

8697 
895i 

8739 
8994 

8782 
9o36 

8824 
9079 

46 

12 

1  3 

3900 

4i56 

3943 
4i98 

3985 
424i 

4028 
4283 

4070 
4326 

4n3 

4368 

47 
46 

r4 
i5 

9121 
9375 

9i63 

94i8 

9206 
9460 

9248 
9602 

9290  9333 
9545  9587 

45 

44 

i4 
i5 

44n 
4667 

4454 
4709 

4496 
4762 

4539 
4794 

458i 
4837 

4624 
4879 

45 
44 

16 

9629 

9672 

97i4 

9757 

9799 

9841 

43 

16 

4922 

4965 

6007 

5o5o 

6092 

5i35 

43 

17 

9884 

9926 

9969 

.  .  ii 

..53 

..96 

42 

ll 

5178 

6220 

5263 

53o5 

5348 

539o 

42 

18 

ic.o5oi38 

0181 

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0265 

o3o8 

o35o 

4i 

18 

5433 

5476 

55i8 

556i 

56o3 

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4i 

19)     0392 

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0477 

0620 

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o6o4 

4o 

J9 

5689 

6731 

6774 

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5859 

59O2 

4o 

201  I0.o5o647 

0689 

0732 

0774 

0816 

o859 

39 

20 

10.066944 

6987 

6029 

6o72 

6n5 

6167 

39 

21 

0901 

o944 

0986 

1028 

1071 

in3 

38 

21 

6200 

6242 

6286 

6328 

6370 

64i3 

38 

22 

n56 

1198 

I24o 

1283 

i325 

i368 

37 

22 

6455 

6498 

654i 

6583 

6626 

6668 

37 

23 

i4io 

i452 

1495 

i537 

i58o 

1622 

36 

23 

6711 

6754 

6796 

6839 

688! 

6924 

36 

24 

i665 

1707 

I749 

1792 

i834 

1877 

35 

24 

6967 

7009 

7062 

7o94 

7i37 

7180 

35 

25 

1919 

1961 

20O4 

2046 

2089 

2l3l 

34 

25 

7222 

7266 

73o8 

735o 

7393 

7435 

34 

26 

2173 

2216 

2258 

2301 

2343 

2386 

33 

26 

7478 

7621 

7563 

7606 

7649 

7691 

33 

27 

9./>r>8 

2470 

25i3 

2555 

2698 

2640 

32 

27 

7734 

7776 

7819 

7862 

7904 

7947 

32 

28 

2682 

2726 

2767 

2810 

2852 

2895 

3i 

28 

799° 

8o32 

8o75 

8117 

8160 

8203 

3i 

2C 

2937 

2979 

3022 

3o64 

3107 

3i49 

3o 

29 

8245 

8288 

833i 

8373 

84i6 

8458 

3o 

3c 

10  053192 

3234 

3276 

33i9 

336i 

34o4 

29 

3o 

io.o685oi 

8544 

8586 

8629 

8672 

87i4 

29 

3i 

3446 

3489 

353i 

3573 

36i6 

3658 

28 

3i 

8767 

8800 

8842 

8885 

8927 

897o 

28 

32 

3701 

3743 

3786 

3828 

387o 

3913 

27 

32 

9oi3 

9o55 

9098 

9141 

9i83 

9226 

27 

33 

3955 

3998 

4o4o 

4o83 

4i25 

4i68 

26 

33 

9269 

93n 

9354 

9397 

9439 

9482 

26 

34 

4210 

4252 

4296 

4337 

438o 

4422 

25 

34 

9626 

9567 

9610 

9652 

9695 

9738 

25 

35 

4465 

4507 

4549 

4592 

4634 

4677 

24 

35 

9780 

9823 

9866 

99o8 

995i 

9994 

24 

36 

4719 

4762 

48o4 

4847 

4889 

4g3i 

23 

36 

10.070036 

0079 

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0164 

0207 

0260 

23 

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4974 

5oi6 

5o59 

5ioi 

5i44 

5i86 

22 

37 

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0378 

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o463 

o5o6 

22 

38 

5229 

6271 

53i4 

5356 

5398 

544  1 

21 

38 

o548 

0691 

o634 

0676 

0719 

0762 

21 

39 

5483 

5526 

5568 

56n 

5653 

5696 

20 

39 

o8o4 

0847 

0890 

0932 

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1018 

20 

4o 

io.o55738 

5781 

5823 

5865 

5908 

595o 

J9 

4o 

10.071060 

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1188 

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1274 

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4i 

5993 

6o35 

6078 

6120 

6i63 

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18 

4i 

i3i6 

i359 

1402 

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i487 

i53o 

18 

42 

6248 

5290 

6333 

6375 

6417 

646o 

I7 

42 

i573 

i6i5 

i658 

1701 

1743 

1786 

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43 

65o2 

6545 

6587 

663o 

6672 

6715 

16 

43 

1829 

1871 

1914 

1967 

1999 

2042 

16 

44 

6757 

6800 

6842 

6885 

6927 

6970 

i5 

44 

2o85 

2128 

2170 

2213 

2256 

2298 

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45 

7012 

7o55 

7097 

7i39 

7182 

7224 

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45 

234l 

2384 

2426 

2469 

2612 

2554 

i4 

46 

7267 

73o9 

7352 

7394 

7437 

7479 

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46 

2697 

2640 

2688 

2726 

2768 

2811 

i3 

47 

7622 

7564 

7607 

7649 

7692 

7734 

12 

47 

2853 

2896 

2939 

2981 

3024 

3067 

12 

48 

7777 

7819 

7862 

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7947 

7989 

1  I 

48 

3no 

3i52 

3196 

3238 

3280 

3323 

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49 

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8o74 

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8i59 

8202 

8244 

10 

49 

3366 

3409 

345  1 

3494 

3537 

3579 

10 

5o 

10.058287 

8329 

8  3  12 

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9 

5o 

10.073622 

3665 

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3793 

3836 

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854i 

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8711 

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8 

5i 

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3921 

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4092 

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52 

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8839 

8881 

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8966 

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7 

52 

4i35 

4178 

4220 

4263 

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4348  7 

53 

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9094 

9i36 

9i79 

9221 

9264 

6 

53 

4391 

4434 

4477 

45i9 

4562 

46o5 

6 

54 
55 

9306 

956i 

9349 
9604 

9391 
9646 

9434 
9689 

9476 
9732 

95i9 

9774 

5 

4 

54 
55 

4648 
4904 

4690 

4947 

4733 
4989 

4776 
5o32 

48i9 
5o75 

486i 
5n8 

5 

4 

56 

9817 

9859 

9902 

9944 

9987 

..29 

3 

56 

5i6o 

52o3 

5246 

6289 

533i 

5374 

3 

5? 

10.060072 

on4 

0167 

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0242 

0284 

2 

57 

54i7 

546o 

55o2 

5545 

5588 

563i 

2 

58 
69 

0327 
o582 

o369 

0624 

o4ra 
0667 

0454 

°7°9 

0497 
0762 

o539 
0794 

I 
0 

58 
59 

5673 
593o 

57i6 
5973 

5759 
6oi5 

58o2 
6o58 

5844  5887 
6101  6i44 

I 
0 

60" 

50" 

40" 

30" 

20" 

10" 

a 

60" 

50' 

40" 

30"  1  20"  1  10" 

ei 

Co-tangent  of  41  Degrees. 

S 

Co-tarigent  of  40  Degrees.    § 

P  P«rt  J  l"   ~"  3"  4"  5//  6"  7//  8"  9" 
z-.i-ari^  4   g   13  17  21  25  go  34  38 

pp  f<  1"  2"  3"  4"  5"  6"  7"  Q"  9" 
ir'}  4   9  13  17  21  26  30  34  38 

7-i 


LOGARITHMIC    SINES. 


c 

Sine  of  50  Degrees. 

d 

Sine  of  51  Degrees. 

s 

0"    |  10"  I  20"  f  30" 

40" 

50" 

i 

0" 

10"  '  20" 

30"^ 

40" 

50'' 

0 

9.884254l4272|4389 

43o7 

4325 

4342 

59 

o 

9.890603 

0620 

o537 

o554 

0671 

0688 

59 

I 

436o 

43784395 

44i3 

443i 

4448 

58 

I 

0606 

0622 

0639 

0666 

o673 

o69o 

58 

2 

4466 

4433i45oi 

4619 

4536 

4554 

67 

2 

0707 

0724 

o74i 

0768 

o775 

0792 

57 

:j 

4672 

4689,4607 

4626 

4642 

466o 

56 

3 

0809 

0826 

o843 

0860 

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o894 

56 

4 

4677 

46954713 

473o 

4748 

4766 

55 

4 

091  1 

O928 

0946 

0962 

°979 

0996 

55 

5    4783 

48oi  43it> 

4636 

4854 

487i 

54 

5 

ioi3 

io3o 

io47 

1064 

1081 

io93 

54 

6,    4889 

4906,4924  494* 

4969 

4977 

53 

6 

in5 

Il32 

1149 

1166 

n83 

1  200 

53 

7 

4994 

6012  5o3o  6o47i5o65 

6082 

52 

7 

1217 

1234 

1261 

1268 

1286 

1302 

62 

8 

6100 

5n85i35 

5i53l5i7o|5i88 

5i 

8 

1  3  19 

i336 

i353 

1370 

i387 

i4o4 

5i 

9J    6206 

62236241 

6268 

627615293 

5o 

9 

1,421 

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i455 

1472 

1489 

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53285346 

5364 

533i 

5399 

49 

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9.891623 

1  54o 

1666 

1673 

1690 

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49 

ii 

54i6 

5434545i 

5469 

5486 

55o4 

48 

ii 

1624 

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1668 

1676 

1692 

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48 

12 

5522 

55395557 

5574 

6692 

6609 

47 

12 

1726 

1743 

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1777 

i794 

1810 

47 

i3 

662-7 

5644,5662 

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6697 

57i4 

46 

i3 

1827 

1  844 

1861 

1878 

1896 

1912 

46 

i4 

5732 

57495767 

5784 

6802 

6819 

45 

i4 

1929 

1946 

I963 

1980 

i996 

2013 

45 

i5 

5837 

5855|5872 

6890 

6907 

6926 

44 

1  5 

2030 

2047 

2064 

2081 

2098 

2116 

44 

16 

6942 

69606977 

6996 

6012 

6o3o 

43 

16 

2l32 

2149 

2166 

2182 

2I99 

2216 

43 

17 

6o47 

6066  6082 

6099 

6117 

6  1  34 

42 

17 

2233 

2260 

2267 

2284 

2300 

23l7 

42 

18 

6162 

6169  6187 

6204 

6222 

6239 

4i 

18 

2334 

235i 

2368 

2385 

24O2 

2419 

4i 

19 

6257 

6274"  6292 

6309 

6327 

6344 

4o 

19!    2435 

2462 

2469 

2486 

25o3 

2620 

4o 

20 

9-886362 

6379'6396 

643i 

6449 

39 

20 

9.892536 

2553 

2570 

2687 

26o4 

2621 

39 

21 

6466 

6484  6601 

6619 

6536 

6554 

38 

21 

2638 

2664 

2671 

2688 

2-706 

2722 

38 

22 

657i 

6588  6606 

6623 

664i 

6658 

37 

22 

2739 

2766 

2772 

2789 

2806 

2823 

37 

23 

6676 

6693  67io 

6728 

6745 

6763 

36 

23 

2839 

2866 

2873 

2890 

2907 

2924 

36 

24 

678o 

6798.6816 

6832 

6860 

6867 

35 

24 

2940 

2967 

2974 

299I 

3oo8 

3024 

35 

25 

6885 

6902  6919 

6937 

6964 

6972 

34 

26 

3o4i 

3o58 

3o75 

3O92 

3io8 

3i25 

34 

26 

6989 

7006  7024 

7°4i 

7069 

7076 

33 

26 

3l42 

3169 

3i76 

3l92 

32O9 

3226 

33 

27 

7093 

7111  7128 

7i46 

7i63 

7180 

32 

27 

3243 

3269 

3276 

3293 

33io 

3327 

32 

»  28 

7198 

721.6  7232 

725o 

7267 

7286 

3i 

28 

3343 

336o 

3377 

3394 

34io 

3427 

3i 

29 

7302 

73197337 

7354 

737i 

7389 

3o 

29 

3444 

346i 

3477 

3494 

35n 

3528 

3o 

3o 

9.887406 

7423  744  1 

7458 

7475 

7493 

29 

3o 

9.893544 

356i 

3578 

3595 

36n 

3628 

29 

3i 

7610 

7628,7645 

7562 

7680 

7597 

28 

3i 

3645 

3662 

3678 

3696 

37I2 

3728 

28 

32 

7614 

7632  7649 

7666 

7684 

7701 

27 

32 

3745 

3762 

3779 

3795 

38i2 

3829 

27 

33 

7718 

7736j7753 

777° 

7787 

7806 

26 

33 

3846 

3862 

3896 

39I2 

3929 

26 

34 

7822 

7839!7857 

7874 

7891 

79°9 

25 

34 

3946 

3963 

3979 

3996 

4oi3 

4029 

25 

35 

7926 

7943  7961 

7978 

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8012 

24 

35 

4o46 

4o63 

4079 

4096 

4n3 

4i3o 

24 

36 

8o3o 

8o47  8064 

8082 

8099 

8116 

23 

36 

4i46 

4i63 

4i8o 

4196 

42i3 

423o 

23 

37 

8i34 

8161  8168 

8186 

82o3  8220 

22 

37 

4246 

4263 

4280 

4296 

43i3 

433o 

22 

38 

8237 

82648272 

8289 

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21 

38 

4346 

4363 

438o 

4396 

44i3 

443o 

21 

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834i 

8358  8375 

8393 

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20 

39 

4446 

4463 

448o 

4496 

45i3 

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9.888444 

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85i3853i 

19 

4o 

9.  894546 

4563 

458o 

4596 

46i3 

4629 

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18 

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4646 

4663 

4679 

4696 

47i3 

4729 

18 

42 

865i 

8669  8686 

87o3 

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17 

42 

4746 

4763 

4779 

4796 

4812 

4829 

17 

43 

8766 

87728789 

8806 

88248841 

16 

43 

4846 

4862 

4879 

4896 

49I2 

4929 

16 

44 

8858 

88768892 

8910 

89278944 

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44 

4945 

4962 

4979 

4995 

5012 

6028 

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45 

896i 

89788996 

9oi3 

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45 

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6062 

6078 

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5ui 

6128 

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46 

9064 

9082  9099 

9116 

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46 

6146 

6161 

6178 

6194 

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47 

9168 

9186  9202 

9219 

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12 

47 

5244 

6261 

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53io 

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12 

48 

9271 

92889306 

9322 

93399366 

II 

48 

5343 

536o 

5377 

5393 

54io 

5426 

I  I 

49 

9374 

9391  94o8 

9426 

94429469 

IO 

49 

5443 

5469 

5476 

5493 

55o9 

6626 

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5o 

9.889477 

94949611 

9628 

96469662 

9 

5o 

9.896642 

6669 

6676 

6692 

6608 

6626 

9 

5i 

9579 

96979614 

963i 

9648  9666 

8 

5i 

564i 

5658 

6676 

6691 

57o8 

5724 

8 

62 

9682 

96999716 

9734 

9761  9768 

7 

52 

6741 

5757 

5774 

6790 

58o7 

5823 

7 

53 

9785 

9802  9819 

9836 

98639871 

6 

53 

584o 

5856 

5873 

6889 

59o6 

6922 

6 

54 

9888 

9906  9922 

9939 

99669973 

5 

54 

5939 

5955 

5972 

6988 

6oo5 

6021 

5 

55 

999° 

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4 

55 

6o38 

6o54 

6071 

6087 

6io4 

6120 

4 

56 

9.890093 

ono  0127 

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0161  0178 

3 

56 

6i37 

6i53 

6170 

6:86 

62o3 

6219 

3 

57 

019^ 

02I2'o23o 

0247 

0264  0281 

2 

57 

6236 

6262 

6269 

6286 

63o2 

63i8 

2 

55 
59 

0298 
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0315^332 

0417,0434 

0349 
o45  1 

o366o383 
o468  o486 

I 

O 

58 
59 

6335 
6433 

635i 
645o 

6368  6384 
64666483 

640016417 
649965i6 

I 
0 

60" 

50"  I  40"   30" 

20"  i  10" 

o 

60" 

50"   40"   30" 

20"   10" 

e 

Co-sirie  of  39  Degrees. 

.n 

Co-sine  of  38  Degrees.     j  § 

p  p   $  1"  2"  3"  4"  5"  6"  7"  8"  9" 

.<  1"  2"  3"  4"  5"  G"  7"  8"  9" 

I  2   3   5   7   9  10  12  14  16 

irt{  2   3   5   7   8  10  12  13  J5 

TANGENT  s. 


I  c  '     Tangent  of  50  Degrees. 

d 

Tangent  of  51  Degrees 

i 

\   \    °" 

10"  j  20" 

30" 

40" 

50" 

s 

0" 

10" 

20" 

30" 

40" 

50" 

J 

olio.  076186 

6229|6272 

63i5 

6358 

64oo 

59 

0 

io.o9i63i 

1674 

1717 

1760 

i8o3 

1  846 

59 

ij    6443 

6486|6529 

657i 

66i4 

6657 

58 

i 

i889 

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!975 

2018 

2061 

2IO4 

58 

2 

3 

6700 
6o56 

6742 
6999 

6785 
7042 

7828 
7o85 

6871 
7127 

69i3 
7170 

57 
56 

2 

3 

2147 
2406 

2I9I 

2449 

2234 
2492 

2277 

2535 

2320 

2578 

2363 
2621 

57 
56 

A 

72l3 

7256 

7298 

734i 

7384 

7427 

55 

4 

2664 

2707 

275o 

2793 

2837 

2880 

55 

5 

747o 

75i2 

7555 

7598 

7641 

7684 

54 

5 

2923 

2966 

3oo9 

3o52 

3o95 

3i3S 

54 

6 

7726 

7769 

7812 

7855 

7897 

794o 

53 

6 

3i8i 

3224 

3267 

33io 

3354 

3397 

53 

7 

7983 

8026 

8069 

8m 

8  1  54 

8i97 

D2 

7 

344o 

3483 

3526 

3569 

36i2 

3655 

52 

8 

8240 

8283 

8325 

8368 

84n 

8454 

5i 

8 

3698 

374i 

3784 

3828 

387i 

39i4 

5i 

9 

8497 

8539 

8582 

8625 

866$ 

8711 

5o 

9 

3957 

4ooo 

4o43 

4o86 

4l29 

4172 

5o 

10 

10.078753 

8796 

8839 

8882 

8925 

8967 

49 

10 

IO.O942l5 

4259 

4302 

4345 

4388 

443  1 

49 

ii 

9010 

9o53 

9096 

9i39 

9181 

9224 

48 

ii 

4474 

45i7 

456o 

46o3 

4647 

469o 

48 

12 

9267 

93io 

9353 

9396 

9438 

948i 

47 

12 

4733 

4776 

48i9 

4862 

49o5 

4948 

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i3 

9524 

9567 

9610 

9652 

9695 

9738 

46 

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5i64 

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0781 

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9867 

9909 

9952 

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45 

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525o 

5293 

5337 

538o 

5423 

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0124 

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0209 

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55o9 

5552 

5595 

5638 

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5725 

44 

16 

0295 

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16 

6768 

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5854 

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5984 

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ll 

6027 

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6i56 

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18 

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1023 

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18 

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39 

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21 

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1666 

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1752 

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38 

21 

7062 

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22 

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1880 

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2009 

2052 

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22 

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23 

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7667 

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24 

2352 

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35 

25 

2609 

2652 

2695 

2738 

2780 

2823 

34 

25 

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8142 

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8271 

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34 

26 

2866 

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2952 

2995 

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33 

26 

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8444 

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853i 

8574 

33 

27 

3i23 

3i6G 

3209 

3252 

3295 

3338 

32 

27 

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8703 

8747 

879o 

8833 

32 

28 

338i 

3424 

3467 

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3552 

3595 

3i 

28 

8876 

89i9 

8963 

9006 

9o49 

9092 

3i 

29 

3638 

368i 

3724 

3767 

38io 

3853 

3o 

29 

9i36 

9J79 

9222 

9265 

93o8 

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3o 

3o 

10-083896 

3938 

398i 

4024 

4067 

4i  10 

29 

3o 

io.o99395 

9438 

948i 

9524 

9568 

96n 

29 

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4282 

4325 

4367 

28 

3i 

9654 

9697 

9?4i 

9?84 

9827 

987o 

28 

32 

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4453 

4496 

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4582 

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27 

32 

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27 

33 

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4882 

26 

33 

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26 

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34 

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35 

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53i2 

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1  384 

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21 

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42 

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2552 

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17 

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44 

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3i58 

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45 

7760 

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7932 

7975 

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45 

3288 

3332 

3375 

34i8 

3462 

35o5 

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46 

8018 

8061 

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8232 

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46 

3548 

3592 

3635 

3678 

3722 

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83i8 

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849o 

12 

47 

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3895 

3938 

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12 

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87o5 

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48 

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49 

8791 

8834 

8877 

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8963 

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10 

49 

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437i 

44r5 

4458 

45oi 

4545 

10 

5o 

10.089049 

9092 

9i35 

9i78 

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9264 

9 

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463i 

4675 

4718 

4761 

48o5 

9 

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9393 

9436 

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8 

5i 

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489i 

4935 

4978 

5021 

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8 

52 

9565 

9608 

965i 

9694 

9737 

978o 

7 

52 

6108 

5i52 

5i95 

5238 

5282 

5325 

7 

53 

9823 

9866 

9909 

9952 

9995 

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6 

53 

5368 

54i2 

5455 

5498 

5542 

5585 

6 

54110.090082 

0125 

0168 

02  1  1 

0254 

0297 

5 

54 

5628 

5672 

57i5 

5759 

58o2 

5845 

5 

55    o34o 

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0426 

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4 

55 

5889 

5932 

5975 

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6062 

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4 

56 

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0727 
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0770 
1028 

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3 

2 

56 
57 

6i49 
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6i92 
6453 

6236 
6496 

6279 
6539 

6322 

6583 

6366 
6626 

•) 

o 

58 

ui4 

n57 

1200 

1243 

1286 

i329 

I 

58 

6669 

67i3 

6756 

6800 

6843 

6886 

i 

59 

1372 

i4i6 

i459 

i5o2 

1  545 

i588 

o 

59 

693o6973 

7017 

7060 

7io3 

7i47 

o 

80" 

50"  |  40" 

30" 

20" 

10" 

60"     50" 

40"   30"   20" 

10" 

d 

Co-tangent  of  39  Degrees. 

S 

Co-tangent  of  38  Degrees. 

•B 

•2 

P  Part*  l"  ~"  3"  4"  5"  6//  7//  8"  9" 
$  4   9  13  17  21  20  30  34  39 

PPartM"  2"  3//  4"  5"  6"  7"  8"  f>" 
r  {  4   9   13  17  22  26  30  3.~>  c9 

LOGARITHMIC    SINES. 


d  1     Sine  of  52  Degrees. 

c 

Sine  of  53  Degrees. 

•S  |    0"    I  10"  1  20' 

30" 

40"   50" 

i 

0" 

10"   20" 

130" 

40" 

50" 

0 

9.8965326649 

663i  6647 

6565658i 
66646680 

6697)6713 

59 

58 

0 

I 

9.9oa349 

2444 

2364 
2460 

238o 
2475 

2396 

249I 

2607 

2428 
2523 

59 

58 

2 

6729(6746 

6762 

6779 

6795 

6812 

57 

2 

2539 

2555 

257I 

2586 

2602 

2618 

57 

3 

6828 

6844 

6861 

6877 

6894 

69io 

56 

3 

2634 

265o 

2666 

2681 

2697 

2713 

56 

4 

6926 

6943 

6959 

6976 

6992 

7°°9 

55 

4 

2729 

2745 

276l 

2776 

2792 

28o8J55 

5 

7025 

7°4i 

7o58 

7074 

7o9o 

7107 

54 

5 

2824 

2840 

2856 

287I 

2887 

29o3 

54 

6 

7123 

7140 

7i56 

7172 

7i89 

7205 

53 

6 

29I9 

2935 

295o 

2966 

2982 

2998 

53 

7 

7222 

7238 

7254 

7271 

7287 

73o3 

52 

7    3oi4 

3o29 

3o45 

3o6i 

3o77 

3093 

52 

g 

7320 

7336 

7353 

7369 

7385 

7402 

5i 

8    3io8 

3i24 

3i4o 

3i56 

3171 

3i87 

5i 

y 

74i8 

7434 

745i 

7467 

7483 

75oo 

5o 

9 

32o3 

32I9 

3235 

325o 

3266 

3282 

5o 

10,9.897516 

7533 

7549 

7565 

7582 

7598 

49 

10 

9.9o3298 

33i3 

3329 

3345 

336i 

3377 

49 

ii 

7614 

763i 

7647 

7663 

7680  7696 

48 

ii 

3392 

34o8 

3424 

344o 

3455 

48 

12 

7712 

7729 

7745  7761 

77787794 

47 

12 

3487 

35o3 

35!8 

3534 

355o 

3566 

47 

i3 

7810 

7827 

7843  7859 

7876  7892 

46 

1  3 

358i 

3597 

36i3 

3629 

3644 

366o 

46 

i4 

79o8 

7924 

7941 

7957 

7973799° 

45 

i4 

3676 

369i 

37o7 

3723 

3739 

3754 

45 

i5 

8006 

8022 

80398055 

8071  8088 

44 

i5 

377o 

3786 

38o2 

38i7 

3833 

3849 

44 

16 

8io4 

8120 

8i368i53 

8i698i85 

43 

16 

3,864 

388o 

3896 

39I2 

3927 

3943 

43 

171    8202 

8218 

8234  825o 

82678283 

42 

17 

3959 

3974 

399o 

4oo6 

4O2I 

4o37 

42 

18 

8299 

83i5 

8332 

8348 

8364 

838i 

4i 

18 

4o53 

4o69 

4o84 

4ioo 

4n6 

4i3i 

4i 

19 

8397 

84i3 

8429  8446 

8462 

4o 

:9 

4i47 

4i63 

4i78 

4i94 

4210 

422D 

4o 

20 

9.898494 

85u 

8527 

8543 

8559 

8576 

39 

20 

9.9o424i 

42  5  7 

4272 

4288 

43o4 

43i9 

39 

21 

8592 

8608 

86248641 

8657 

8673 

38 

21 

4335 

435i 

4366 

4382 

4398 

44i3 

38 

22 

8689  8706 

8722 

8738 

8754 

877o 

3? 

22 

4429 

4445 

446o 

44?6 

4492 

45o7 

37 

23 

8787 

88o3 

8819 

8835 

8852 

8868 

36 

23 

4523 

4539 

4554 

4570 

4586 

46oi 

36 

24 

8884 

8900 

8916 

8933 

8949 

8965 

35 

24 

46i7 

4632 

4648 

4664 

4679 

4695 

35 

25 

898i 

8997 

9014 

9o3o 

9o46 

9o62 

34 

25 

4711 

4726 

4742 

4757 

4773 

4789 

34 

26    9078 

9095 

9111 

9I27 

9i43 

9i59 

33 

26 

48o4 

4820 

4836 

485i 

4867 

4882 

33 

27   9176 

9192 

9208 

9224 

9240 

9256 

32 

27 

4898 

49i4 

4929 

4945 

496o 

4976 

32 

28   9273 

9289 

93o5 

932I 

9337 

9354 

3i 

28 

4992 

5007 

5o23 

5o38 

5o54 

5o7o 

3i 

291   9370 

9386 

9402 

<;4i8 

9434 

945o 

3o 

29 

5o85 

5ioi 

5n6 

5i32 

5i48 

5i6! 

3o 

809.899467 

9483 

9499 

95i5 

953! 

9547 

29 

3o 

9.9o5i79 

5i94 

5210 

5225 

524i 

52t'f 

2O 

3i 

9564 

9580 

9596 

9612 

9628 

9644 

28 

3i 

5272 

5288 

53o3 

53i9 

5334 

535o 

2o 

32 

966o 

9677 

96V3 

9709 

9725 

974i 

27 

32 

5366 

538i 

5397 

54i2 

5428 

5443 

27 

33 

9757 

9773 

9789 

9806 

9822 

9838 

26 

33 

5459 

5474 

549o 

55o6 

552i 

5537 

26 

34 
35 

9854 
995i 

9870  g886  9902 
9967(9983  9999 

9918 

9935 

25 

24 

34 
35 

5552 
5645 

5568 
566i 

5583 
5676 

5599 
5692 

56i4 
57o8 

563o 
5723 

25 

24 

36 

9.9ooo47 

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0112 

0128 

23 

36 

5739 

5754 

577o 

5785 

58oi 

58i6 

23 

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0176  OI92 

0208 

0224 

22 

37 

5832 

5847 

5863 

5878 

5894 

5909 

22 

38 

0240 

02*16 

0272  O289 

o3o5 

0321 

21 

38 

5925 

594o 

5956 

597i 

5987 

6002 

21 

39 

o337o353 

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o4i7 

2O 

39 

6018 

6o33 

6o49 

6o64 

6080 

6095 

2O 

4o 

9.900433(0449 

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o497 

o5i3 

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4o 

9.9o6ni 

6126 

6142 

6i57 

6i73 

6188 

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4i 

0629 

o545 

o562  0578 

o594 

0610 

18 

4i 

6204 

62I9 

6235 

6260 

6265 

6281 

1  8 

42 

o6«6 

0642 

o658  0674 

o69o 

o7o6 

17 

42 

6296 

63i2 

6327 

6343 

6358 

6374 

17 

43 

0722 

0738 

0754  0770 

o786 

0802 

16 

43 

6389 

64o5 

6420 

6436 

645  1 

6466  1  6 

44 

0818 

o834 

o85oo866 

0882 

o898 

i5 

44 

648a 

6497 

65i3 

6528 

6544 

6559 

i5 

45 

0914 

0980 

o946|o962 

o978 

o994 

i4 

45 

6575 

659o 

66o5 

6621 

6636 

6652 

i4 

46 

fOIO 

1026 

IO42 

io58 

io74 

IO9O 

i3 

46 

6667 

6683 

6698 

67i3 

6729 

6744 

i3 

47 

/1  06 

1122 

n38 

n54 

1170 

1186 

12 

47 

6760 

6775 

679i 

6806 

6821 

6837 

12 

48 

12O2 

1218 

1234 

I25o 

1266 

1282 

II 

48 

6852 

6868 

6883 

6898 

69i4 

6929 

II 

49 

1298 

i3i4 

i33o 

1  346 

i362 

i378 

10 

49 

6945 

696o 

6975 

699i 

7006 

7O22 

10 

56 

q  qoi394 

i4io 

i4a6 

1  442 

i458 

1  4?4 

9 

5o 

9.9o7o37 

7052 

7o68 

7o83 

7o99 

7n4 

9 

5i 

1490 

i5o5 

l52I 

i537 

r553 

i569 

8 

5i 

7I29 

7i45 

7i6o 

7i75 

7i9i 

7206 

8 

52 

i585 

1601 

1617 

i633 

i649 

i665 

7 

52 

7222 

7237 

7252 

7268 

7283 

7298 

7 

53 

1681 

1697 

I7i3 

1729 

i745 

1760 

6 

53 

73i4 

7329 

7344 

736o 

7375 

739i 

6 

54 

1776 

1792 

1808 

1824 

i84o 

i856 

5 

54 

7406 

7421 

7437 

7452 

7467 

7483 

5 

55 

1872 

1888 

i9o4 

1920 

i936 

i95i 

4 

55 

7498 

75i3 

7529 

7544 

7559 

7575 

4 

56 

1967 

i983 

i999 

2Ol5 

2031 

2047 

3 

56 

759° 

76o5 

762I 

7636 

765i 

7667 

3 

57 

2o63 

2079 

2095 

2110 

2126 

2l42 

2 

57 

7682 

7697 

77i3 

7728 

7743 

7759 

2 

58 

2i58 

2174 

2I90 

2206 

2222 

2238 

I 

58 

7774 

7789 

78o5 

7820 

7835 

785i 

I 

59 

2253 

2269 

2285 

2301 

2317 

2333 

O 

59 

7866 

7881 

7896 

79I2 

7927 

794a 

O 

60" 

50" 

40" 

30"  |  20" 

10" 

a 

60" 

50' 

40" 

30"   20" 

10" 

d 

Co-sine  of  37 

Degrees. 

Co-sine  of  36  Degrees.     |  3 

(  I"  °"  3"  4" 
P.  Part  ^  2356 

5"  f5''  7"  8'  9" 
8  10  11  13  15 

I'.PartJ  2  ~3   5   6   8   9   11  12  14  ' 

LOGARITHMIC     TANGENTS. 


n 

Tangent  of  52  Degrees. 

d 

Tangent  of  53  Degrees. 

3 

()''     |  10"   20"   30" 

40"   50" 

% 

0"      10"   20"   So"   40"   50" 

0 

10.  107190 

723Z 

7277 

7320 

7364 

74o7 

59 

o 

10.122885 

2929 

2973 

3017 

3o6i 

3io5 

59 

I 

745  1 

7494 

7537 

758i 

762/ 

7668 

58 

i 

3i48 

3i92 

3236 

3280 

3324 

3368 

58 

2 

7711 

7754 

7798 

784i 

7885 

79.28 

57 

2 

34xi 

3455 

3499 

3543 

3587 

363o 

57 

^ 

7972 

8oi5 

8o58 

8102 

8i45 

8189 

56 

3 

3674 

3718 

3762 

38o6 

385o 

3893 

56 

4 

8232 

8275 

8319 

8362 

84o6 

8449 

55 

4 

3937 

398i 

402D 

4o69 

4u3 

4i57 

55 

£ 

8493 

8536 

8579 

8623 

8666 

8710 

54 

f; 

4200 

4244 

4288 

4332 

4376 

4420 

54 

6 

8753 

8797 

884o 

8884 

8927 

8970 

53 

G 

4463 

45o7 

455i 

4595 

4639 

4683 

53 

7 

9014 

9o57 

9101 

9i44 

9i88 

923l 

52 

7 

4727 

4770 

48i4 

4858 

49O2 

4946 

52 

8 

9275 

93i8 

936i 

94o5 

9448 

9492 

5i 

8 

4990 

5o34 

5o77 

5l2I 

5i65 

5209 

5i 

9 

9535 

9579 

0,622 

9666 

97°9 

9753 

5o 

9 

5253 

5297 

534i 

5385 

5428 

5472 

5o 

10 

10.  109796 

984o 

9883 

9926 

997° 

..i3 

49 

10 

10.  1255:6 

556o 

56o4 

5648 

5692 

5736 

/ 

1  1 

to.  i  10057 

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oi87 

0231 

0274 

4» 

i  ] 

578o 

5823 

5867 

59u 

5955 

5999 

12 

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o36i 

o4o5 

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0492 

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47 

12 

6o43 

6o87 

6i3i 

6i75i62i9 

6262 

47 

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46 

q 

63o6 

635o 

6394 

6438 

6482 

6526 

46 

U 

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45 

4 

657o 

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6789 

45 

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5 

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16 

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G 

7°97 

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7i85 

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43 

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1622 

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^T-1 

7 

736o 

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7448 

7492 

7536 

758o 

42 

18 

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7756 

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20 

10.  112406 

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2623 

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10.  128i5i 

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39 

21 

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21 

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85o3 

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38 

22 

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37 

22 

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23 

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25 

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34 

25 

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969i 

34 

20 

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33 

26 

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9955 

33 

27 

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4279 

4322 

4366 

4409 

4453 

32 

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28 

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29 

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29 

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8562 

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865o 

8695 

O 

60"    |  50"   40"   30"   20" 

10" 

K 

60"      50" 

40"   30"   *0" 

10" 

d 

Co-tangent  of  37  Degrees. 

§ 

Co-tangent  of  36  Degrees. 

& 

T»  "P^i-  j  1"  2"  3"  4"  5"  6"  7"  8"  9" 

P  Port  j  *"  2"  3"  4"  5"  6"  7"  S"  9" 

n|  4  9  13  17  22  26  31  35  39 

f.  rart.  -j  4  9  13  18  22  o6  31  35  40 

.s 

Sine  of  54  Degrees. 

a 

Sine  of  55  Degrees.      j 

•1 

0"     10" 

20"   30"   40"  |  50"  | 

3     0"     10" 

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60" 

50" 

40"  |  30"   20"   10"   „. 

|    60"     50"   40"   30"   20"   10"   fl. 

Co-sine  of  35  Degrees.     2 

!     Co-sine  of  34  Degrees.     2 

P  PJVt5  l"  ~"  3"  4"  5"  6"  T  8"  9"   ,.  „  A  1"  2"  3"  4"  5"  6"  7"  6"  9" 

)  2   3   5 

G   8   9  11  12  14  :     "l\  1   3   4   6   7   9  10  12  13 

LoGAKIlHMIfJ      T  A  N  G  F  N  T  S. 


d 

Tangent  of  54  Degrees. 

e     Tangent  of  55  Degrees. 

3 

0"      10-' 

20" 

30" 

40"  |  50" 

0" 

10" 

20" 

30" 

40" 

50" 

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o876 

0922 

u967 

o 

(30" 

50" 

40" 

30" 

20" 

10" 

. 

GO"     50" 

40" 

30" 

20" 

10"  1  d 

Co-tangent  of  35  Degrees. 

9 

Co-tangent  of  34  Degrees.    1  3 

p  p   (  1"  2"  3"  4"  5"  6"  7"  8"  9" 

f  -[it  nil  3"  4"  5"  Q"  7"  g"  9" 

I  4   9  13  18  22  27  31  36  40 

P.  Part  J  5   g  14  ig  03  o  /  32  36  41 

80 


LOGARITHMIC    SINES. 


I  — 

1  £ 

Sine  of  56  Degrees. 

.5 

Sine  of  57  Degrees.     )   | 

2 

0' 

10" 

20" 

30'' 

40"  |  50" 

2 

0"    j  10" 

20"  [  30" 

40" 

50' 

0 

9.918574 

8588 

86o3 

8617 

863ij8645 

59 

o 

9.923591 

36o5 

36l9 

3632 

3646 

366o 

59 

i 

8659 

8674 

8688 

8702 

8716 

873o 

58 

i 

3673 

3687 

37oi 

37i4 

372S 

3742 

58 

2 

8745 

8759 

8773 

8787 

8801 

88i5 

57 

'2 

3755 

3769 

3783 

3796 

38io 

3824 

57 

3 

883o 

8844 

8858 

8872 

8886 

8900 

56 

3 

3837 

385i 

3865 

3878 

3892 

39o6l56 

4 

89i5 

8929 

8943 

8957 

8971 

8985 

55 

4J    3919 

3933 

3946 

3960 

3974 

3987i55 

5 

9000 

9014 

9028 

9042 

go56 

9070 

54 

51    4ooi 

4oi5 

4028 

4042 

4o55 

4069 

54 

U 

9085 

9099 

9113 

9127 

9141 

9i55 

53 

6 

4o83 

4096 

4no 

4124 

4i37 

4i5i 

5'3 

7 

9169 

9184 

9!98 

9212 

9226 

9240 

52 

7 

4i64 

4i78 

4192 

42o5 

4219 

4232 

5a 

8 

9254 

9268 

9282 

9297 

93u 

9325 

5i 

8 

4246 

4260 

4273 

4287 

43oo 

43i4 

5i 

9 

9339 

9353 

9367 

938i 

9395 

94io 

5o 

9 

4328 

434i 

4355 

4368 

4382 

4396 

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10 

9.919424 

9438 

9452 

9466 

9480 

9494 

49 

IO 

9.924409 

4423 

4436 

445o 

4464 

4477 

49 

ii 

95o8 

9522 

9537 

955i 

9565 

9579 

48 

ii 

4491 

45o4 

45i8 

453i 

4545 

4559 

48 

12 

9593 

9607 

9621 

9635 

9649 

9663 

47 

12 

4572 

4586 

4599 

46i3 

4626 

464o 

47 

i3 

9677 

9692 

9706 

9720 

9734 

9748 

46 

i3 

4654 

4667 

468i 

4694 

47o8 

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46 

i4 

9762 

9776 

979° 

9804 

9818 

9832 

45 

i4 

4735 

4748 

4762 

4776 

4789 

48o3 

45 

i5 

98^6 

9860 

9875 

9889 

9903 

9917 

44 

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48-i  6 

483o 

4843 

4857 

487o 

4884 

44 

16 

993i 

9945 

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9973 

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.  .  .  i 

43 

16 

4897 

49n 

4924 

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4952 

4965 

43 

i? 

9.9200l5 

0029 

oo43 

0057 

0071 

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42 

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4979 

4992 

5oo6 

5019 

5o33 

5o46 

42 

18 

oo99 

on3 

0127 

oi4i 

oi56 

0170 

4i 

18 

5o6o 

5o73 

5o87 

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5n4 

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4i 

J9 

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0198 

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0226 

0240 

0254 

4o 

J9 

5i4i 

5i54 

5i68 

5i8i 

5i95 

5208 

4o 

20 

9.920268 

0282 

0296 

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39 

20 

9.925222 

5235 

5249 

5262 

5276 

5289 

39 

21 

o352 

o366 

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o394 

o4o8 

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38 

21 

53o3 

53i6 

533o 

5343 

5357 

537o 

38 

22 

o436 

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o478 

0492 

o5o6 

37 

22 

5384 

5397 

54n 

5424 

5438 

545i 

37 

23 

o52O 

o534 

o548 

o562 

0576 

oSgo 

36 

23 

5465 

5478 

549i 

55o5 

55i8 

5532 

36 

24 

0604 

0618 

o632 

o646 

0660 

0674 

35 

24 

5545 

5559 

5572 

5586 

5599 

56i3 

35 

25 

0688 

0702 

0716 

0730 

0744 

o758 

34 

25 

5626 

564o 

5653 

5667 

568o 

5693 

34 

261    0772 

0786 

0800 

0814 

0828 

0842 

33 

26 

57o7 

5720 

5734 

5747 

576i 

5774 

33 

27|    0866 

0869 

o883  0897 

0911 

0925 

32 

27 

5788 

58oi 

58i4 

5828 

584i 

5855 

32 

28 

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0967 

0981 

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1009 

3! 

28 

5868 

5882 

5895 

5908 

5922 

5935 

3! 

29 

1023 

1037 

io5i 

io65 

1079 

1093 

3o 

29 

5949 

5962 

5976 

5989 

60O2 

6016 

3o 

3o 

9.921  107 

II2I 

1  1  34 

n48 

1162 

1176 

29 

3o 

9.926029 

6o43 

6o56 

6069 

6o83 

6096 

29 

3i 

1190 

1204 

1218 

1232 

1246 

1260 

28 

3i 

6110 

6i23 

6i36 

6i5o 

6i63 

6i77 

28 

32 

1274 

1288 

1302 

i3i5 

1329 

1  343 

27 

32 

6190 

6203 

62I7 

623o 

6244 

6257 

27 

'  33 

i357 

1371 

i385 

i399 

i4i3 

1427 

26 

33 

6270 

6284 

629-7 

63n 

6324 

6337 

26 

34 

i44i 

i455 

i468  1482 

1496 

i5io 

25 

34 

635i 

6364 

6377 

639i 

64o4 

64i8 

25 

35 

i524 

i538 

i552  i566 

i58o 

i593 

24 

35 

643  1 

6444 

6458 

6471 

6484 

6498 

24 

36 

1607 

1621 

i635li649 

i663 

1677 

23 

36 

65ii 

6525 

6538 

655i 

6565 

6578 

23 

37 

1691 

1704 

1718 

1732 

1746 

1760 

22 

37 

659i 

66o5 

6618 

663i 

6645 

6658 

22 

38 

1  774 

1788 

1802 

i8i5 

1829 

i843 

21 

38 

667i 

6685 

6698 

67n 

6725 

6738 

21 

39 

1857 

1871 

i885 

!899 

!9I2 

1926 

2O 

39 

675i 

6765 

6778 

679i 

68o5 

6818 

2O 

4o 

9.921940 

1954 

1968 

I982 

I995 

2009 

I9 

4o 

9.926831 

6845 

6858 

687i 

6885 

6898 

J9 

4i 

2023 

2037 

2o5i 

2o65 

2079 

2092 

18 

4.i 

6911 

6925 

6938 

695i 

6965 

6978 

18 

42 

2106 

2I2O 

2134 

2148 

2l62 

2175 

17 

42 

6991 

7oo5 

-7018 

7o3i 

7o44 

7o58 

17 

43 

2189 

2203 

2217 

223l 

2244 

2258 

16 

43 

7o7i 

7084 

7o98 

7in 

7I24 

7i38 

16 

44 

2272 

2286 

2300 

23i3 

2327 

234l 

i5 

44 

7i5i 

7164 

7i77 

7191 

7204 

72I7 

i5 

45!    2355 

2369 

2383 

2396  2410 

2424 

i4 

45 

723l 

7244 

7257 

7270 

-7284 

7297 

i4 

AQ 

2438 

2452 

2465 

2479 

2493 

2507 

i3 

46 

73io 

7324 

7337 

735o 

7363 

7377 

i3 

47 

252O 

2534 

2548 

2562 

2576 

2589 

12 

47 

739o 

74o3 

74i6 

743o 

7443 

7456 

12 

48 

2603 

2617 

263i 

2644 

2658 

2672 

II 

48 

747o 

7483 

7496 

75o9 

7523 

7536 

I  I 

49 

2686 

2700 

2713 

2727 

274l 

2755 

IO 

49 

7549 

7562 

7576 

7589 

76o2 

76i5 

10 

5o 

9.922768 

2782 

2796 

2810 

2823 

2837 

9 

5o 

9.927629 

7642 

7655 

7668 

768i 

7695 

9 

5i 

285i 

2865 

2878 

2892 

2906 

2920 

8 

5i 

77o8 

7721 

7734 

7748 

7761 

7774 

8 

52 

2933 

2947 

2961 

2975 

2988 

3002 

7 

52 

7787 

7801 

78i4 

7827 

784o 

7853 

7 

53 

3oi6 

3o3o 

3o43 

3o57 

3071 

3o84 

6 

53 

7867 

788o 

7893 

79o6 

7920 

7933 

6 

54 

3o98 

3lI2 

3i26 

3i3g 

3  1  53 

3i67 

5 

54 

7946 

7959 

7972 

7986 

7999 

8012 

5 

55 

3i8i 

3i94 

32o8 

3222 

3235 

3249 

4 

55 

8o25 

8o38 

8o52 

8o65 

8o78 

8o9i 

4 

56 

3263 

3277 

3290 

33o433i8 

333i 

3 

56 

8io4 

8118 

8i3i 

8i44 

8i57 

8170!  3 

57    3345 

3359 

3372 

338634oo 

34i4 

2 

57 

8i83 

8197 

8210 

8223 

8236 

8249  2 

58 

3427 

344  1 

3455 

3468  3482 

3496 

I 

58 

826382-76 

8289 

83o2 

83i5 

8328  i 

59 

35o9 

3523 

353  7 

355o3564 

3578 

0 

59 

8342J8355 

8368 

838i 

8394 

84o7j  ° 

60" 

50" 

40" 

30"   20" 

10" 

C 

60'    j  50" 

40" 

30"   20"   10"  j 

Co-sine  of  33  Degrees. 

3 

Co-sine  of  32  Degrees.    1  3 

p  „  .<  1"  2"  3"  4"  5"  6"  7"  8"  9" 

v  „  ,  (  1"  2"  3"  4"  5"  6"  7"  8"  9" 

irt$  1   3  4  6   7   8  10  11  13 

irt\  1   3   4   5   7   8   9  11  12 

LOGARITHMIC    TANGENTS 


d 

Tangent  of  56  Degrees. 

_g 

Tangent  of  57  Degrees. 

53 

"0"" 

10" 

20" 

30" 

40" 

50" 

3 

0" 

10" 

20'' 

30" 

40"  |  50" 

ojio.  171018 

io58 

no3 

n49 

H94 

I24o 

59 

0 

10.187483 

7529 

7575 

762I 

7667 

77i3 

59 

I 

1285 

i33i 

i376 

1421 

i467 

l5l2 

58 

I 

7759 

78o5 

785i 

7898 

7944 

799° 

58 

ii 

i558 

i6o3 

1649 

1694 

i739 

i785 

57 

2 

8o36 

8082 

8128 

8i74 

8220 

82^7 

5? 

3 

i.83o 

i876 

1921 

J967 

2OI2 

2o58 

56 

3 

83i3 

8359 

84o5 

845i 

8497  8543 

56 

4 

2103 

2149 

2I94 

2239 

2285 

233o 

55 

4 

859o 

8636 

8682 

8728 

87748820 

55 

5|     2376 

2421 

2467 

25l2 

2558 

26o3 

54 

5 

8866 

89i3 

8959 

9oo5 

9o5i  9o97 

54 

6 

2649 

2694 

2740 

2785 

283i 

2876 

53 

6 

9i43 

9190 

9236 

9282 

9328  o3y4 

53 

7 

2922 

2g67 

3oi3 

3o58 

3io4 

3i49 

52 

7 

9420 

9467 

95i3 

9559|96o5 

965i 

52 

8 

3195 

324o 

3286 

333i 

3377 

3422 

5i 

8 

9698 

9744 

979° 

9836 

9882 

9929 

5! 

9 

3468 

35i3 

3559 

36o4 

365o 

3695 

5o 

9 

9975 

.  .21 

..67 

.n3 

.  160 

.206 

5o 

10 

10.173741 

3786 

3832 

3877 

39a3 

3968 

49 

10 

10.  190252 

0298 

o344 

o39i 

o437 

o483 

49 

1  1 

4014 

4o6o 

4io5 

4i5i 

4i96 

4242 

48 

II 

o529 

0576 

0622 

0668 

0714 

0760 

48 

12 

4287 

4333 

4378 

4424 

4469 

45i5 

47 

12 

0807 

o853 

0899 

o945 

0992 

io38 

47 

i3 

456i 

46o6 

4652 

4697 

4743 

4788 

46 

i3 

1084 

n3o 

1177 

1223 

1269 

i3i5 

46 

U 

4834 

488o 

4925 

497i 

5oi6 

5o62 

45 

i4 

i362 

i4o8 

i454 

i5oi 

1  547 

i593 

45 

i5 

5107 

5i53 

5i99 

5244 

529O 

5335 

44 

i,5 

i639 

1686 

I732 

i778 

1824 

1871 

44 

16 

538i 

5427 

5472 

55i8 

5563 

56o9 

43 

16 

1917 

J963 

2O10 

2o56 

2IO2 

2l49 

43 

I? 

5655 

57oo 

5746 

579i 

5837 

5883 

42 

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2195 

2241 

2287 

2334 

2380 

2426 

42 

18 

5928 

5974 

6019 

6o65 

6111 

6i56 

4i 

18 

2473 

25l9 

2565 

2612 

2658 

2704 

4i 

19 

6202 

6247 

6293 

6339 

6384 

643o 

4o 

J9 

2751 

2797 

2843 

289o 

2936 

2982 

4o 

20 

10.  176476 

652i 

6567 

66i3 

6658 

67o4 

39 

20 

10.  193029 

3o75 

3l2I 

3i68 

32l4 

3260 

39 

21 

6749 

6795 

684i 

6886 

6932 

6978 

38 

21 

3307 

3353 

3399 

3446 

3492 

3538 

38 

22 

7023 

7o69 

7116 

7i6o 

7206 

7252 

3? 

22 

3585 

363i 

3678 

3724 

3770 

38i7 

37 

23 

7297 

7343 

7389 

7434 

748o 

7526 

36 

23 

3863 

39o9 

3956 

4OO2 

4049 

4095 

36 

24 

757i 

76i7 

7663 

77o8 

7754 

78oo 

35 

24 

4i4i 

4i88 

4234 

42&I 

4327 

4373 

35 

25 

7846 

789i 

7937 

7983 

8028 

8o74 

34 

25 

4420 

4466 

45i3 

4559 

46o5 

4652 

34 

26 

8120 

8i65 

8211 

8257 

83o3 

8348 

33 

26 

4698 

4745 

479i 

4837 

4884 

4g3o 

33 

37 

8394 

844o 

8485 

853i 

8577 

8623 

32 

27 

4977 

5o23 

5070 

5n6 

5i62 

5209 

32 

28 

8668 

87i4 

8760 

88o5 

885i 

8897 

3i 

28 

5255 

53o2 

5348 

5395 

544i 

5487 

3i 

29 

8943 

8988 

9o34 

9o8o 

9126 

9171 

3o 

29 

5534 

558o 

5627 

5673 

5720 

5766 

3o 

3o 

10.  179217 

9263 

93o9 

9354 

9400 

9446 

29 

3o 

IO.  I958l3 

5859 

59o6 

5952 

5999 

6o45 

29 

3i 

9492 

9537 

9583 

9629 

9675 

9720 

28 

3i 

6091 

6i38 

6184 

623i 

6277 

6324 

28 

32 

9766 

9812 

9858 

99o4 

9949 

9995 

27 

32 

63?o 

64i7 

6463 

65io 

6556 

66o3 

27 

33 

10.  i8oo4i 

oo87 

Ol32 

oi78 

0224 

0270 

26 

33 

6649 

6696 

6742 

6789 

6835 

6882 

26 

34 

o3i6 

o36i 

0407 

o453 

o499 

o545 

25 

34 

6928 

6975 

7021 

7o68 

7114 

7161 

26 

35 

o5go 

o636 

0682 

0728 

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o8i9 

24 

35 

7208 

7254 

73oi 

7347 

7394 

744o 

24 

36 

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ioo3 

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io94 

23 

36 

7487 

7533 

758o 

-7626 

7673 

7719 

23 

37 

n4o 

1186 

1232 

12-78 

i323 

i369 

22 

*7 

7766 

78i3 

7859 

7906 

7952 

7999 

22 

38 

i4i5 

i46i 

1507 

i553 

i598 

1  644 

21 

38 

8o45 

8o92 

8i38 

8i85 

8232 

8278 

21 

39 

1690 

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1782 

1828 

i874 

i9i9 

2O 

39 

8325 

837i 

84i8 

8465 

85n 

8558 

20 

4o 

10.181965 

2011 

2057 

2103 

2l49 

2195 

:9 

4o 

10.  198604 

865i 

8697 

8744 

8791 

8837 

'9 

4i 

224l 

2286 

2332 

2378 

2424 

2470 

18 

4i 

8884 

893o 

8977 

9024 

9070 

9117 

18 

42 

25i6 

2562 

2608 

2653 

2699 

2745 

*7 

42 

9164 

92IO 

9257 

93o3 

935o 

9397 

ll 

43 

2791 

2837 

2883 

2929 

2975 

3021 

16 

43 

9443 

949o 

9537 

9583 

9630 

9676 

16 

44 

3o67 

3lI2 

3i58 

32o4 

325o 

3296 

i5 

44 

9723 

977° 

98i6 

9863 

9910 

9956 

i5 

45 

3342 

3388 

3434 

3480 

3526 

3572 

i4 

45 

IO.2OOOO3 

oo5o 

oo96 

oi43 

0190 

0236 

i4 

46 

36i8 

3664 

37o9 

3755 

38oi 

3847 

i3 

46 

0283 

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o376 

o423 

0470 

o5i6 

i3 

47 

3893 

3939 

3985 

4o3i 

4o77 

4l23 

12 

47 

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0610 

o656 

0703 

0750 

0796 

12 

48 

4i69 

42i5 

4261 

43o7 

4353 

4399 

I  I 

48 

o843 

o89o 

o937 

o983 

io3o 

1077 

II 

49 

4445 

4491 

4537 

4583 

4629 

4674 

10 

49 

1123 

1170 

I2I7 

1263 

i3io 

i357 

10 

5o 

10.184720 

4766 

4812 

4858 

4904 

495o 

9 

5o 

IO.2Ol4o4 

i45o 

i497 

1  544 

iSgi 

i637 

9 

5i 

4996 

5o42 

5o88 

5:34 

5i8o 

5226 

8 

5i 

1684 

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1777 

1824 

1871 

1918 

8 

52 

5272 

53i8 

5364 

54io 

5456 

55o2 

7 

52 

i964 

2OII 

2o58 

2105 

2l5l 

2I98 

7 

53 

5548 

5594 

564o 

5686 

5732  5778 

6 

53 

2245 

2292 

2338 

2385 

2432 

2479 

6 

54 

5824 

587i 

59i7 

5963 

6009  6o55 

5 

54 

2526 

2572 

2619 

2666 

2713 

3759 

5 

55 

6101 

6i47 

6i93 

6239 

6285633i 

4 

55 

2806 

2853 

2900 

2947 

2993 

3o4o 

4 

56 

6377 

6423 

6469 

65i5 

656i  66o7 

3 

56 

3087 

3i34 

3i8i 

3227 

3274 

332i 

3 

57 

6653 

6699 

6745 

679i 

6837  6884 

2 

57 

3368 

34i5 

346i 

35o8 

3555 

36o2 

3 

58 

693o 

6976 

7022 

7o68 

7u4  7i6o 

I 

58 

3649 

3696 

3742 

3789 

3836 

3883 

X 

1? 

72O6 

7252 

7298 

7344 

739o  7437 

0 

59 

393o 

3977 

4023 

4070 

4117 

4i64 

o 

~~ 

GO"      50" 

40" 

30"   20"   10" 

a 

60"      50"  |  40"   30" 

20" 

10" 

Co-tangent  of  33  Degrees. 

2 

Co-tangent  of  32  Degrees. 

a 

p  p   41"  2"  3"  4"  5"  6"  7"  8"  9" 

(  1"  2"  3"  4"  5"  6"  7"  8"  9" 

(  5   9  14  18  23  27  32  37  41 

irt}  5   9  14  19  23  28  33  37  42 

LOGARITHMIC    SINES. 


Jj 

Sine  of  58  Degrees. 

d 

s3 

Sine  of  59  Degrees. 

7  "~* 

8 

0" 

10" 

20"   30"  |  40" 

50" 

s 

0" 

10" 

20" 

30" 

40" 

50" 

0 

9.  928420 

8434 

844784608473 

8486 

59 

o  9.933066 

3078 

Sogi 

3  10^ 

3n6 

3i29 

59 

I 

8499 

85i3 

8526  8539  8552 

8565 

58 

i 

3i4i 

3i54 

3i67 

3i79 

3i92 

32o5 

58 

2 

8578 

859i 

86o586i8!863i 

8644 

57 

2 

3217 

323o 

3243 

3255 

3268 

328o 

57 

3 

8657 

8670 

8683  8696  8710 

8723 

J56 

J 

3293 

33o6 

33i8 

333i 

3344 

3356 

56 

4 

8736 

8749 

8762  87758788 

8801 

55 

L 

3369 

338i 

3394|34o7 

34i9 

3432 

55 

5 

88i5 

8828l884i  8854  8867 

888o|54 

i 

3445 

3457 

3470 

3482 

3495 

35o8 

54 

6 

8893 

89o689i989338946 

8959l53 

6 

3520 

3533 

3545 

3558 

357i 

35b3 

53 

7 

8972 

8985 

8998  9oii 

9024 

9o37 

52 

7 

3596 

36o8 

362i 

3633 

3646 

3659 

52 

8 

9o5o 

9o63 

9077  9o9o9io3 

9116 

5i 

8 

3671 

3684 

3696 

37o9 

3?22 

3734 

5i 

9 

9129 

9i42  9i55 

9i68 

9181 

9i94 

5o 

9 

3747 

3759 

3772 

3784 

3797 

38io 

5o 

10 

9.929207 

922O!9233 

9247 

9260 

9273 

49 

IO 

9.933822 

3835 

3847 

386o 

3872 

3885 

49 

ii 

9286 

92999312 

9325 

9338 

935i 

48 

ii 

3898|39io 

3923 

3935 

3948 

396o 

48 

12 

9364 

93771939° 

94o3 

94169429 

47 

12 

3973 

3985 

3998 

4on 

4o23 

4o36 

47 

i3 

9442 

9455;9468 

9482 

9495 

95o8 

46 

i3 

4648 

4o6i 

4073 

4o86 

4o98 

4m 

46 

i4 

9521 

9534:9547 

956o 

9573 

9586 

45 

i4 

4l23 

4i36 

4i48 

4i6i 

4i74 

4i86 

45 

i5 

9599 

96l2 

9625 

9638 

965i 

9664 

44 

i5 

4199 

4211 

4224 

4236 

4249 

4261 

44 

16 

9677 

969o 

97°3 

97169729 

9742 

43 

16 

4274 

4286 

4299 

43n 

4324 

4336 

43 

17 

9755 

9768978i 

9794  9807 

9820 

42 

17 

4349 

436i 

4374 

4386 

4399 

44" 

42 

18 

9833 

9846  9859  9872  9885 

9898 

4i 

18 

4424 

4436 

4449 

446! 

4474 

4486 

4i 

'9 

9911 

9924 

9937  995o9963 

9976 

4o 

19 

4499 

45n 

4524 

4536 

4549 

456i 

4o 

20 

9.929989 

.  .  .2 

.  .  i5 

..28 

.  .4i 

..54 

39 

20 

9.934574 

4586 

4599 

46  n 

4624 

4636 

39 

21 

oo8o'oo93 

0106 

on9 

Ol32 

38 

21 

4049 

466i 

4674 

4686 

4699 

4711 

38 

22 

oi45 

oi58 

0171 

0184 

oi97 

O2IO 

37 

22 

4723 

4736 

4748 

4761 

4773 

4786 

37 

23 

O223 

O236  0249 

0262 

0274 

0287 

36 

23 

4798 

48" 

4823 

4836 

4848 

486i 

36 

24 

o3oo 

o3i3 

o326 

o339 

o352 

o365 

35 

24 

4873 

4885 

4898 

49io 

4923 

4935 

35 

25 

o378 

o39i 

o4o4 

0417 

o43o 

o443 

34 

25 

4948 

496o 

4973 

4985 

4997 

5oio 

34 

26 

o456 

o469 

0482 

0495 

o5o7 

O52O 

33 

26 

5O22 

5o35 

5o47 

5o6o 

5o72 

5o84 

33 

27 

o533 

o546 

o559 

0572 

o585 

o598 

32 

27 

5o97 

5io9 

5l22 

5i34 

5i47 

5i59 

32 

28 

0611 

0624  o637 

o65o 

o663 

o675 

3i 

28 

5171 

5i84 

5i96 

5209 

5221 

5234 

3i 

29 

0688 

o7oi 

0714 

0727 

0740 

o753 

3o 

29 

5246 

5258 

0271 

5283 

5296 

53o8 

3o 

3o 

9.930766 

°779  °792 

o8o4 

0817 

o83o 

29 

3o 

9.93532o 

5333 

5345 

5358 

537o 

5382 

29 

3i 

o843 

o856  o869 

0882 

o895 

0908 

28 

3i 

5395 

54o7 

5420 

5432 

5444 

5457 

28 

32 

O92I 

o933  o946 

o959 

°972 

o985 

27 

32 

5469 

5482 

5494 

55o6 

55i9 

553i 

27 

33 

o998 

IOII 

1024 

io36 

io49 

1062 

26 

33 

5543 

5556 

5568 

558i 

5593 

56o5 

26 

34 

1075 

1088 

I  101 

in4 

1127 

n39 

25 

34 

56i8 

563o 

5642 

5655 

5667 

5679 

25 

35 

Il52 

n65 

1178 

1191 

I2O4 

1217 

24 

35 

5692 

57o4 

57i7 

5729 

574i 

5754 

24 

36 

I229 

1242 

1255 

1268 

I28l 

I294 

23 

36 

5766 

5778 

579i 

58o3 

58i5 

5828 

23 

37 

i3o6 

i3i9 

i332 

1  345 

i358 

1371 

22 

37 

584o 

5852 

5865 

5877 

5889 

5902 

22 

38 

i383 

i396 

i4o9 

1422 

i435 

1  448 

21 

38 

59i4 

5926 

5939 

595i 

5963 

5976 

21 

39 

i46o 

i473 

i486 

1  499 

l5l2 

i525 

2O 

39 

5988 

6000 

6oi3 

6o25 

6o37 

6o5o 

2O 

4o 

9.93i537 

i55o 

i563 

i576 

i589 

1601 

'9 

4o 

9.936o62 

6074 

6o87 

6o99 

6m 

6124 

I9 

4i 

1614 

1627 

i64o 

i653 

1666 

1678 

18 

4i 

6i36 

6i48 

6161 

6i73 

6i85 

6i98 

73 

.  42 

i69i 

1704 

1717 

1730 

1742 

i755 

17 

42 

6210 

6222 

6234 

6259 

627I 

'7 

43 

1768 

1781 

i794 

1806 

i8i9 

i832 

16 

43 

6284 

6296 

63o8 

6320 

6333 

6345 

16 

44 

i845 

i857 

1870 

i883 

i896  I9o9 

i5 

44 

6357 

6370 

6382 

6394 

64o6 

64i9 

i5 

45 

I92I 

i934 

i947 

1960 

i972  i985 

i4 

45 

643i 

6443 

6456 

6468 

648o 

649s 

i4 

46 

i998 

2OII 

2024 

2o36 

2o49  2062 

i3 

46 

65o5  65i7 

6529 

6542 

6554 

6566 

i3 

47 

2075 

2087 

2IOO 

2Il3 

2126  2i38 

12 

47 

6578 

659i 

66o3 

66i5 

6627 

664o 

12 

48 

2l5l 

2164 

2177 

2189 

2202  22  1  5 

11 

43 

6652 

6664 

6676 

6689 

67oi 

67i3 

II 

49 

2228 

2240 

2253 

2266 

2279  229I 

IO 

49 

6725 

6738 

675o 

6762 

6774 

6787 

10 

5o 

9.932304 

23l7 

2329 

2342 

2355 

2368 

9 

5o 

j.936799 

6811 

6823 

6836 

6848 

6860 

9 

5i 

238o 

2393 

2406 

24l9 

243i 

2444 

8 

5i 

6872 

6884 

6897 

69o9 

692I 

6933 

8 

52 

2457 

2469  2482 

2495 

25o8 

2520 

7 

52 

6946 

6958 

697o 

6982 

6994 

7oo7 

7 

53 

2533 

25462558 

2.71 

2584!2597 

6 

53 

7oi9 

7o3i 

7o43 

7o56 

7o68 

7o8o 

6 

54 

2609 

2622  2635 

2647 

26602673 

5 

54 

7°92 

7104 

7n7 

7I29 

7i4i 

7i53 

5 

55 

2685 

2698 

2711 

2724 

2736l2749 

4 

55 

7i65 

7178 

7I90 

7202 

7214 

•7226 

4 

56 

2762 

2774:2787 

28002812^825 

3 

56 

7238 

725i 

7263 

7275 

7287 

7299 

3 

57 

2838 

285o2863 

2876 

2888  29OI 

2 

57 

73l2 

7324 

7336 

7348 

736o 

7372 

2 

58 

2914 

2926  2939 

2952 

2964  2977 

I 

58 

7385 

7397 

74o9 

742I 

7433 

7446 

I 

59 

2990 

3oo2  3oi5 

3028 

3o4o  3o53 

O 

59 

7458 

7470 

7482 

7494 

75o6 

75i8 

O 

60" 

50" 

40"   30"   20"   10" 

d 

60"     50" 

40"   30" 

20"   10" 

Co-sine  of  31  Degrees. 

§ 

Co-sine  of  30  Degrees. 

P  Part*  l"  ~"  3" 

4"  5"  6"  7"  8"  9" 

„  ,,  ,  (  1"  2"  3"  4"  5"  6"  7"  8"  9"  i 

1  i  1   3   4 

5   6   8   9  10  12      "")  12   4567   9  10  11' 

LOGARITHMIC    TANGENTS. 


o  o 
o  o 


c 

i  — 
Tangent  of  58  Degrees. 

j3 

Tangent  of  59  Degrees. 

% 

0"     |  10'-'   20"  [  30"   40"   50" 

s 

0"     |  10"   20" 

30" 

40" 

JjO« 

0 

10.20421  1 

4268 

43o4 

435i 

4398 

4445 

r 

o 

IO.22I226 

1274 

l322 

i369 

1417 

i465 

59 

i 

4492 

4539 

4586 

4633 

4679 

4726 

58 

I 

l5l2 

i56o 

l6o8 

i656 

1703 

i75i 

58 

2 

4773 

482o|4867|49i4 

496i 

5oo8 

57 

2 

I799 

1  846  i894 

I942 

1990 

2037 

57 

3 

5o54 

5roi 

5i48 

5i95 

5242 

5289 

56 

3 

2o85 

2i33 

2181 

2228 

2276 

2324 

56 

4 

5336 

5383 

543o 

5477 

5524 

557o 

55 

4 

2372 

24l9 

2467 

25i5 

2563 

26lO 

55 

5 

56i7 

5664 

5711 

5758 

58o5 

5852 

54 

5 

2658 

2706 

2754 

2801 

2849 

2897 

54 

6 

5899 

5946 

5993 

6o4o 

6087 

6i34 

53 

6 

2945 

2993 

3o4o 

3o88 

3i36 

3i84 

53 

7 

6181 

6227 

6274 

632i 

6368 

64i5 

52 

7 

3232 

3279 

3327 

3375 

3423 

347i 

52 

8 

6462 

65o9 

6556 

66o3 

665o 

6697 

5i 

8 

35i8 

3566 

36i4 

3662 

3710 

3757 

5i 

9 

6744 

679i 

6838 

6885 

6.982 

6979 

5o 

9 

38o5 

3853 

3901 

3949 

3997 

4o44 

5o 

10 

10  .207026 

7o73 

7120 

7167 

7214 

7261 

49 

10 

ro.224o92 

4i4o 

4i88 

4236 

4284 

4332 

49 

ii 

73o8 

7355 

7402 

?449 

7496 

7543 

48 

1  1 

4379 

4427 

4475 

4523 

457i 

46i9 

48 

12 

i3 

759° 

7872 

7637 
79i9 

7684 
7966 

773i 
8oi3 

7778 
8060 

7825 
8107 

47 
46 

12 

i3 

46674714 

4954  5oo2 

4762 
5o5o 

48io 
5o98 

4858 
5i45 

49o6 
5i93 

47 

46 

i4 

8i54 

8201 

8248 

8295 

8342 

8389 

45 

i4 

524i 

5289 

5337 

5385 

5433 

548  1 

45 

i5 

8437 

8484 

853i 

8578 

8625 

8672 

44 

i5 

5529 

5577 

5625 

5672 

5720 

5768 

44 

16 

8719 

8766 

88i3 

8860 

89o7 

8954 

43 

16 

58i6 

5864 

5912 

5960 

6008 

6o56 

43 

'7 

9001 

9o48 

9o95 

9i43 

9i9o 

9237 

42 

*7 

6io4 

6i52 

6200 

62^8 

6296 

6344 

42 

id 

9284 

933i 

9378 

9425 

9472 

95i9 

4i 

18 

6392 

644o 

6488 

6535 

6583 

663i 

4i 

J9 

9566 

96i4 

966i 

0708 

9755 

98o2 

4o 

r9 

6679 

6727 

6775 

6823 

6871 

69i9 

4o 

20 

io.2o9849 

9896 

9.943 

999  l 

..38 

..85 

3  9 

20 

10.226*567 

7Oi5 

7o63 

7111 

7i59 

7207 

39 

21 

IO.2IOI32 

oi79 

0226 

0273 

0321 

o368 

38 

21 

7255 

73o3 

735i 

7399 

7447 

7495 

38 

22 

o4i5 

0462 

o5o9 

o556 

o6o3 

o65i 

37 

22 

7543 

759i 

7639 

7688 

7736 

7784 

37 

23 

0698 

o745 

0792 

o839 

0886 

o934 

36 

23 

7832 

7880 

-7928 

7976 

8024 

8072 

36 

24 

o98i 

1028 

1075 

1122 

1170 

1217 

35 

24 

8120 

8168 

8216 

8264 

83i2 

836o 

35 

25 

1264 

i3n 

i358 

i4o5 

i453 

i5oo 

34 

25 

84o8 

8456 

85o4 

8552 

8601 

8649 

34 

26 

i547 

i594 

i64i 

i689 

i736 

i783 

33 

26 

8697 

8745 

8793 

884i 

8889 

8937 

33 

27 

i83o 

1878 

I925 

I972 

2OI9 

2066 

32 

27 

8985 

9o33 

9o8i 

9i3o 

9178 

9226 

32 

28 

2Il4 

2161 

2208 

2255 

23o3 

235o 

3i 

28 

9274 

9322 

937° 

94:8 

9466 

95i5 

3i 

29 

2397 

2444 

2492 

2539 

2586 

2633 

3o 

29 

9563 

9611 

9659 

9707 

9755 

98o3 

3o 

3o 

TO.  2I268l 

2728 

2775 

2822 

2870 

29i7 

29 

3o 

10.229852 

9900 

9948 

9996 

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.  .92 

29 

3i 

2964 

3012 

3o59 

3io6 

3i53 

3201 

28 

3i 

io.23oi4o 

0189 

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32 

3248 

3295 

3343 

339o 

3437 

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27 

32 

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27 

33 

3532 

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3626 

3674 

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3768 

26 

33 

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26 

34 

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3863 

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3958 

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25 

34 

1008 

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25 

35 

4ioo 

4i47 

4:94 

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4289 

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24 

35 

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1  345 

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1490 

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24 

36 

4384 

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4478 

4526 

4573 

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23 

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1779 

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23 

37 

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22 

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1924 

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2021 

2069 

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22 

38 

4952 

4999 

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5o94 

5i4i 

5i89 

21 

38 

2166 

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2262 

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2359 

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21 

39 

5236 

5284 

533i 

5378 

5426 

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39 

2455 

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2552 

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10.232745 

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2938 

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18 

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3228 

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42 

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6137 

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17 

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3373 

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35i8 

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43 

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6422 

6469 

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44 

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6849 

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44 

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45 

6944 

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45 

4i95 

4244 

4292 

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4389 

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46 

7229 

7276 

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4486 

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4582 

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4679 

4728 

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47 

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12 

47 

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12 

48 

7799 

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794i 

7989 

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48 

5o67 

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5212 

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49 

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8274 

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10 

49 

5357 

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10 

5o 

io.2i8369 

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9 

5o 

io.235648 

5696 

5745 

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9 

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8654 

8702 

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8797 

8845 

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6181 

62 

894o 

8987 

9o35 

9083 

9i3o 

9178 

7 

52 

623o 

6278 

6327 

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6424 

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7 

53,    9225 

9273 

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9368 

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6 

53 

652i 

6569 

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67i5 

6763 

6 

54 

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5 

54 

6812 

6860 

69o9 

6957 

7006 

7o55 

5 

55 

9797 

9844 

9892 

9939 

9987 

..35 

4 

55 

7io3 

7i52 

720O 

7249 

7297 

7346 

4 

56 

10.220082 

oi3o 

0178 

0225 

0273 

O32I 

3 

56 

7394 

7443 

7492 

754o 

7589 

7637 

o 

57 

o368 

o4i6 

o463 

o5i  i 

o559 

0606 

2 

57 

7686 

7734 

7783 

7832 

7880 

7929 

2 

58 
59 

o654 
o94o 

0702 

o749 
io36 

0797 

io83 

o845 
n3i 

0892 

1179 

I 

O 

58 
59 

7977 
8269 

8026 

83i8 

8o758i23 
8366|84i5 

8172 
8463 

8220 
85i2 

I 
O 

0>' 

50"   40" 

130" 

20" 

10" 

Q 

60"    1  50"   40"   30"  |  20"  j  10" 

dj 

Co-tangent  of  3  1  Degrees. 

s 

Co-tangent  of  30  Degrees. 

§ 

"prarJ1"  ~"  3//  4//  5"  6"  7"  8"  9"  Ii  PPart*1"  2"  3"  4//  5//  G"  7"  8"  9" 

{  5   9  14  19  24  28  33  38  43  !  1-Jcirt}  5   JQ  14  19  24  29  34  39  43 

SINES. 


4 

Sine  of  60  Degrees. 

.£ 

Sine  of  6  1  Degrees. 

fl 

0" 

10"  |  20"  1  30" 

40"  |  50" 

* 

0"    |  10" 

20" 

30" 

40" 

50" 

c 

\;.93753j 

7543 

7555 

7567 

7579 

759i 

59 

O 

9.94i8i9 

i83i 

1843 

i854 

1866 

1878 

59 

I 

7604 

76l6 

7628 

764c 

7652 

7664 

58 

I 

i889 

1901 

I9i3 

192^ 

1936 

i948 

58 

2 

7676 

7689 

7701 

77i3 

7725 

7737 

57 

2 

i959 

1971 

i983 

1994 

2006 

2017 

57 

3 

7749 

7761 

7773 

7786 

7798 

7810 

56 

3 

2O29 

2O52 

206^ 

2076 

2087 

56 

4 

7822 

7834 

7846 

7858 

7870 

7883 

55 

4 

2099 

2III 

2122 

2i34|2i46 

2l57 

55 

5 

7895 

79°7 

79i9 

793i 

7943 

7955 

54 

5 

2169 

2l8o 

2I92 

22O4  22l5 

2227 

54 

6 

7967 

7979 

7992 

8oo4 

8016 

8028 

53 

6 

2239 

2250 

2262 

2273 

2285 

2297 

53 

7 

8o4o 

8o52 

8o64 

8076 

8088 

8100 

52 

7 

23o8 

2320 

233l 

2343 

2355 

2366 

5a 

8 

8n3 

8i25 

8i37 

8i49 

8161 

8i73 

5i 

8 

2378 

2390 

2401 

24i3 

242^ 

2436 

5! 

9    8i85 

8i97 

82O9  8221 

82338245 

5o 

9 

2448 

2459 

2471 

2482 

2494 

25o6 

5o 

109.938258 

82-70 

8282  820.4 

83o6 

83i8 

49 

10 

9.9425i7 

2529 

254o 

2552 

2563 

2575 

49 

ii 

833o 

8342 

83548366 

8378 

839o 

48 

ii 

2587 

2598 

2610 

2621 

2633 

2645 

48 

12 

8402 

84i4 

84268439 

845i 

8463 

47 

12 

2656 

2668 

2679 

2691 

2702 

27l4 

47 

i3 

8475 

8487 

84998611 

8523 

8535 

46 

i3 

2726 

2737 

2749 

2760 

2772 

2783 

46 

i4 

8547 

8559 

857i|8583 

8595 

86o7 

45 

i4 

2795 

2806 

2818 

283o 

2841 

2853 

45 

i5 

8619 

863i 

8643  8655 

8667 

8679 

44 

i5 

2864 

2876 

2887 

2899 

2910 

2922 

44 

16 

8691 

87o3 

8715  8727 

8739 

875i 

43 

16 

293^ 

2945 

2957 

2968 

2980 

299I 

43 

17 

8763 

8776 

8788  8800 

8812 

8824 

42 

17 

3oo3 

3oi4 

3026 

3o37 

3o49 

3o6o 

42 

18 

8836 

8848 

88608872 

8884 

8896 

4i 

18 

3072 

3o83 

3o95 

3107 

3n8 

3i3o 

4i 

19 

8908 

8920 

8932  8944 

8956 

8968 

4o 

i9 

3i4i 

3i53 

3i64 

3i76 

3i87 

3i99 

4o 

20 

9.938980 

8992 

9oo4  9oi6 

9028 

9o4o 

39 

20 

9.9432io 

3222 

3233 

3245 

3256 

3268 

39 

21 

9062 

9o64 

9O76  9o87 

9099 

9ni 

38 

21 

3279 

329I 

33o2 

33i4 

3325 

3337 

38 

22 

9123 

9i35 

9i479i59 

9171 

9i83 

37 

22 

3348 

336o 

337i 

3383 

3394 

34o6 

37 

23 

9195 

9207 

92i9923i 

9243 

9255 

36 

23 

34i7 

3429 

344o 

3452 

3463 

3475 

36 

24 

9267 

9279 

929i  93o3 

93i5 

9327 

35 

24 

3486 

3498 

35o9 

352i 

3532 

3543 

35 

25 

9339 

935i 

9363  9375 

9387 

9399 

34 

25    3555 

3566 

3578 

3589 

36oi 

36i2 

34 

26 

9410 

9422 

9434  9446 

9458 

0.470 

33 

26 

3624 

3635 

3647 

3658 

3670 

368i 

33 

27 

9482 

9494 

95o6  95i8 

953o 

9542 

32 

27 

3693 

37o4 

37i5 

3727 

3739 

375o 

32 

28 

9554 

9566 

9578959o 

96oi 

96i3 

3i 

28 

376i 

3773 

3784 

3796 

38o7 

38i8 

3i 

29 

9625 

9637 

9649  9661 

9673 

9685 

3o 

29 

383o 

384i 

3853 

3864 

3876 

3887 

3o 

809.939697 

97°9 

972i  9733 

9744 

9756 

29 

3o 

9.943899 

39io 

392I 

3933 

3944 

3956 

29 

3i 

9768 

978o 

9792  98o4 

98i6 

9828 

28 

3i 

3967 

3978 

399o  4ooi 

4oi3 

4024 

28 

32 

9840 

9852 

98639875 

9887 

9899 

27 

q 

4o36 

4o47 

4o58  4o7o 

4o8i 

4o93L 

27 

33 

9911 

9923 

9935  9947 

9959 

997° 

26 

33 

4io4 

4n5 

4127 

4i38 

4i5o 

4i6i 

26 

34 

9982  9994 

...6 

..18 

..3o 

..42 

25 

34 

4172 

4i84 

4195 

4207 

4218 

4229 

25 

35 

9,94oo54 

oo65 

0077  oo89 

OIOI 

on3 

24 

35 

4241 

4252 

4264 

4275 

4286 

4298 

24 

36 

OI25 

oi37 

0148  0160 

OI72 

oi84 

23 

36 

43o9 

4321 

4332 

4343 

4355 

4366 

23 

37 

0196 

0208 

O22O  O23l 

0243 

0255 

22 

37 

4377 

4389 

44oo 

44i2 

4423 

4434 

22 

38 

0267 

0279 

O29I 

o3o3 

o3i4 

o326 

21 

38 

4446 

4457 

4468 

448o 

449i 

45o3 

21 

39 

o338 

o35o 

o362  0374 

o385 

o397 

2O 

39 

45i4 

4525 

4537 

4548 

4559 

457i 

2O 

4o 

9.94o4o9 

0421 

0433  0445 

o456 

o468 

9 

4o 

9.944582 

4593 

46o5 

46i6 

4627 

4639 

I9 

4i 

o48o 

0492 

o5o4  o5i6 

o527 

o539 

8 

4i 

465o 

466i 

4673 

4684 

4696 

47o7 

18 

42 

o55i 

o563 

o575o586 

o598 

0610 

7 

42 

47i8 

473o 

474i 

4752 

4764 

4775 

17 

43 

0622 

o634 

o645 

0667 

0669 

0681 

6 

43 

4786 

4798 

4809 

4820 

483i 

4843 

16 

44 

o693 

0704 

0716  0728 

o74o 

O752 

5 

44 

4854 

4865 

4877 

4888 

4899 

4911 

i5 

45 

0763 

o775 

07870799 

0811 

0822 

4 

45 

4922 

4933 

4945 

4956 

4967 

4979 

i4 

46 

o834 

o846 

o858 

0870 

0881 

0893 

3 

46 

4990 

5ooi 

5oi3 

5024 

5o35 

5o46 

i3 

47 

o9o5 

o9i7 

O928 

0940 

0962 

0964 

2 

47 

5o58 

5o69 

5o8o 

5092 

5io3 

5u4 

12 

48 

o975 

o987 

0999 

ion 

1023 

io34 

I 

48 

5i25 

5i37 

5i48 

5i5g 

5i7i 

5i82 

II 

49 

io46 

io58 

1070 

1081 

1093 

no5 

O 

49 

5i93 

52o4 

52i6 

5227 

5238 

525o 

IO 

5o 

9.94IH7 

1128 

n4o 

Il52 

n64 

n75 

9 

5o 

9.945261 

5272 

5283 

5295 

53o6 

53I7 

9 

5i 

1187 

1199 

121  I 

1222 

1234 

1246 

8 

5i 

5328 

534o 

535i 

5362 

5374 

5385 

8 

52 

1258 

I269 

I28l 

1293 

i3o4 

i3i6 

7 

52 

5396 

5407 

54i9 

543o 

544i 

5452 

7 

53 

i328 

i34o 

i35i 

i363 

i375 

i387 

6 

53 

5464 

5475 

5486 

5497 

55o9 

552O 

6 

54 

i398 

i4io 

1422 

i433 

i445 

i457 

5 

54 

553i 

5542 

5554 

5565 

5576 

5587 

5 

55 

i469 

i48o 

l492 

i5o4 

i5i5 

l527 

4 

55 

5598 

56io 

562i 

5632 

5643 

5655 

4 

56 

i539 

i55o 

i562 

i574 

i586 

159-7 

3 

56 

5666 

5677 

5688 

5700 

57n 

5722 

3 

57 

i6o9 

1621 

632 

1  644 

i656 

i667 

2 

57 

5733 

5744 

5756 

5767 

5778 

5789 

2 

58 

l679 

i69i 

702 

i7i4 

I726 

i738 

I 

58 

58oo58i2 

5823 

5834 

5845 

585? 

I 

59 

i749 

1761 

773 

i784 

i796 

1808 

O 

59 

5868  5879  5890 

5901  5gi3  5924 

O 

60"     50" 

40" 

30"   20"   10" 

3 

60"     50"   40"   30"   20"   1(X' 

. 

Co-sine  of  29  Degrees. 

Co-sine  of  28  Degrees. 

a 

p  p   C  1"  2"  3"  4"  5"  6"  7"  8"  0" 
in\l   2   4  5   6   7   8  10  11 

p     \  I"  2"  3"  4"  5"  6"  7"  8"  9" 

LOGARITHMIC    TANGENTS. 


S5 


.s 

5? 

Tangent  of  60  Degrees. 

.s' 
s 

Tangent  of  61  Degrees. 

0" 

10" 

20"   30" 

40"   50" 

0" 

10" 

20" 

30" 

40"   50" 

o 

10.  23856i 

8609 

86583707 

8755 

88o4 

59 

0 

10.256248 

6298 

6347 

6397 

6447 

6496 

59 

I 

8852 

8901 

895o 

8998 

9047 

9096 

58 

I 

6546 

6596 

6645 

6695 

6745 

6794 

58 

2;    oi44 

9:93 

9242 

9290 

9339 

9388 

57 

2 

6844 

6894 

6944 

6993 

7043 

7o93 

57 

3 

9436 

9485 

9534 

9582 

963i 

9680 

56 

3 

7142 

719.2 

7242 

729i 

7341 

739i 

56 

4 

9728 

9777 

9826 

9874 

9923 

9972 

55 

4 

744i 

7490 

754o 

759o 

7640 

7689 

55 

5 

10.240021 

0069 

0118 

0167 

O2l5 

0264 

54 

5 

7739 

7789 

7839 

7888 

7938 

7988  54 

6 

7 

o3i3 
o6o5 

o362 
o654 

o4io 

0703 

0459 
0752 

o5o8 
0800 

o557 
0849 

53 

52 

6 
7 

8o38 
8336 

8087 
8386 

8i37 
8436 

8187 
8486 

8237 
8535 

8286 

8585 

53 

52 

8 

0898 

0947 

0995 

io44 

io93 

n4a 

5i 

8 

8635 

8685 

8735 

8784 

8834 

8884 

5i 

9 

1190 

1239 

1288 

i337 

i385 

i434 

5o 

9 

8934 

8984 

9o33 

9o83 

9i33 

9i83 

5o 

10 

io.24i483 

i532 

i58i 

1629 

1678 

1.727 

49 

10 

10.259233 

9283 

9332 

9382 

9432 

9482 

49 

1  1 

1776 

i825 

i873 

1922 

1971 

2O2O 

48 

1  1 

9532 

9582 

9632 

968i 

973i 

9781 

48 

12 

2069 

2118 

2166 

22l5 

2264 

23l3 

47 

12 

983i 

9881 

993i 

998i 

..3i 

..80 

47 

i3 

2362 

2411 

2459 

25o8 

2557 

2606 

46 

i3 

io.26oi3o 

0180 

O23O 

0280 

o33o 

o38o 

46 

1.4 

2655 

2704 

2753 

2801 

285o 

2899 

45 

i4 

o43o 

o48o 

o53o 

o58o 

o629 

0679 

45 

i5 

2948 

2997 

3o46 

3095 

3i43 

3i92 

44 

i5 

0729 

0779 

o829 

08.79 

O929 

0979 

44 

16 

324l 

3290 

3339 

3388 

3437 

3486 

43 

16 

1029 

1079 

II29 

n79 

I229 

1279 

43 

17 

3535 

3584 

3632 

368i 

373o 

3779 

42 

'7 

1329 

i379 

l429 

1479 

l529 

i579 

42 

18     8828 

3877 

3926 

3975 

4024 

4o73 

4i 

18 

1629 

1679 

I729 

1779 

l829 

1879 

4i 

19 

4l22 

4171 

4220 

4269 

43i8 

4366 

4o 

X9 

1929 

1979 

2029 

2079 

2I29 

2179 

4o 

20 

10.  2444i5 

4464 

45i3 

4562 

46zi 

466o 

39 

20 

10.262229 

2279 

2329 

2379 

2429 

2479 

39 

21 

4709 

47^8 

48o7 

4856 

49o5 

4954 

38 

21 

2529 

2579 

2629 

2679 

2729 

2779 

38 

22 

5oo3 

5o52 

5ioi 

5i5o 

5199 

5248 

37 

22 

2829 

2879 

2929 

2979 

3o29 

3o79 

37 

23 

5297 

5346 

5395 

5444 

5493 

5542 

36 

23 

3i3o 

3i8o 

3230 

3280 

333o 

338o 

36 

24 

559i 

564o 

5689 

5738 

5787 

5836 

35 

24 

343o 

348o 

353o 

358o 

363o 

368i 

35 

25 

5885 

5934 

5984 

6o33 

6082 

6i3i 

34 

25 

373i 

378i 

383i 

388  1 

393i 

3981 

34 

26 

6180 

6229 

6278 

6327 

6376 

6425 

33 

26 

4o3i 

4082 

4l32 

4182 

4232 

4282 

33 

27 

6474 

6523 

6572 

6621 

6670 

6720 

32 

27 

4332 

4382 

4433 

4483 

4533 

4583 

32 

28 

6769 

6818 

6867 

6916 

6965 

7014 

3i 

28 

4633 

4683 

4734 

4784 

4834 

4884 

3i 

29 

7063 

7112 

7161 

7211 

7260 

73o9 

3o 

2  9 

4934 

4985 

5o35 

5o85 

5i35 

5  1  85 

3o 

3o 

ic.247358 

7407 

7456 

75o5 

7554 

7604 

29 

3o 

io.265236 

5286 

5336 

5386 

5436 

5487 

29 

3i 

7653 

7702 

775i 

7800 

7849 

7899 

28 

3i 

5537 

5587 

5637 

5688 

5738 

5788 

28 

32 

7948 

7997 

8o46 

8o95 

8i44 

8194 

27 

32 

5838 

5889 

5939 

5989 

6o39 

6090 

27 

33 

8243 

8292 

834i 

8390 

8439 

8489 

26 

33 

6i4o 

6190 

6240 

629I 

634i 

639i 

26 

34 

8538 

8587 

8636 

8685 

8735 

8784 

25 

34 

6442 

6492 

6542 

6592 

6643 

6693 

25 

35 

8833 

8882 

8931 

8981 

9o3o 

9079 

24 

35 

6743 

6794 

6844 

6894 

6945 

6995 

24 

36 

9128 

9178 

9227 

9276 

9325 

9375 

23 

36 

7045 

7096 

7i46 

7i96 

7247 

7297 

23 

3? 

9424 

9473 

9522 

9572 

9621 

9670 

22 

37 

7347 

7398 

7448 

7498 

7549 

7599 

22 

38 

9719 

9769 

9818 

9867 

9916 

9966 

21 

38 

7649 

77oo 

775o 

7800 

785i 

79oi 

21 

39 

io.25ooi5 

0064 

on4 

oi63 

O2I2 

0261 

2O 

39 

7952 

8002 

8o52 

8io3 

8i53 

8204 

20 

4o 

io.25o3n 

o36o  0409 

0459 

o5o8 

o557 

X9 

4o 

10.268254 

83o4 

8355 

84o5 

8456 

85o6 

r9 

4i 

0607 

o656 

0705 

o755 

o8o4 

o853 

18 

4i 

8556 

86o7 

8657 

8708 

8758 

88o9 

18 

42 

0903 

0952 

1001 

io5i 

IIOO 

n49 

17 

42 

8859 

89o9 

896o 

9oio 

9o6i 

9m 

«7 

43 

1199 

1248 

1297 

1  347 

i396 

i445 

16 

43 

9l62 

92I2 

9263 

93i3 

9364 

94i4 

16 

44 

i495 

1  544 

i594 

i643 

1692 

1742 

i5 

44 

9465 

95i5 

9566 

9616 

9667 

9717 

i5 

45 

1791 

i84o 

1890 

i939 

1989 

2038 

i4 

45 

9767 

98i8 

9868 

99i9 

997° 

.  .20 

i4 

46 

2087 

2137 

2186 

2236 

2285 

2335 

r3 

46 

10.270071 

OI2I 

0172 

0222 

0273 

o323 

i3 

47 

2384 

2433 

2483 

2532 

2582 

263i 

12 

47 

o374 

O424 

0475 

0525 

0576 

0626 

12 

48 

2681 

2730 

2779 

2829 

2878 

2928 

II 

48 

0677 

0728 

0778 

o829 

o879 

0930 

II 

49 

2977 

3027 

3076 

3i26 

3i75 

3225 

10 

49 

o98o 

io3i 

1082 

Il32 

n83 

1233 

10 

5o 

10.253274 

3324 

3373 

3423 

3472 

352i 

9 

5o 

10.271284 

i335 

i385 

i436 

i486 

i537 

9 

5i 

3571 

3620 

3670 

3719 

3769 

38i8 

8 

5i 

i588 

i638 

i689 

i739 

I79'0 

i84i 

8 

52 

3868 

39i8 

3967 

4017 

4o66 

4n6 

7 

52 

i89i 

I942 

I993 

2o43 

2O94  2l45 

7 

53 

4i65 

42i5 

4264 

43i4 

4363 

44i3 

6 

53 

2I95 

2246 

2297 

2347 

2398  2449 

6 

54 

4462 

45i2 

456i 

46n 

466i 

4710 

5 

54 

2499 

255o 

26OI 

265i 

2702  275  1 

5 

55 

476o 

4809 

4859 

4908 

4958 

5oo8 

4 

55 

2803 

<,:854 

29o5 

2955 

3oo6;3o57 

4 

56 

5o57 

5107 

5i56 

52o6 

5256 

53o5 

3 

56 

3io8 

3i58 

32O9 

3260 

33io!336i 

3 

$7 

5355 

54o4 

5454 

55o4 

5553 

56o3 

2 

57 

34i2 

3463 

35i3 

3564 

36i5|3666 

2 

58 
59 

5652 
595o 

5702 
6000 

5752 
6049 

58oi 
6099 

585i 
6149 

59oi 
6i98 

I 
0 

58 
59 

37i6 

4O2I 

3767 
4072 

38i8 

4l22 

3869[39i9'397o 
4173^224)4276 

O 

60" 

50" 

40" 

no"  20" 

10" 

d 

00" 

50" 

40" 

3<>'  |  20"   10" 

c 

Co-tangent  of  29  Degrees. 

a 

Co-tangent  of  28  Degrees. 

1 

P  Part$  l"  ~"  3"  4//  5"  6"  7"  8"  9// 

}  5  10  15  20  25  29  34  39  44 

.(  1"  2"  3"  4"  5"  6"  7"  8"  9" 
irt\  5  10  15  20  25  30  35  40  45 

86 


LOGARITHMIC    SINES. 


I 

Sine  of  62  Degrees. 

d 

Sine  of  63  Degrees. 

m 

0" 

10" 

20"'   30"   40" 

50" 

s 

0" 

10" 

20"   30"  |  40" 

50" 

0 

9.945935 

5946595-7,59695980 

5991 

59 

0 

9.94988i 

9892 

9902 

99i3  992/ 

9935 

59 

6002 

601360246036,6047 

6o58 

58 

I 

9945 

9956 

9967 

9977 

9988 

9999 

58 

2 

6069 

6080  6092  6io3  61  1^ 

6i25 

57 

t 

9.950010 

OO2O 

oo3i 

0042 

OOD2 

oo63 

57 

C 

6i3( 

6147615961706181 

6192 

56 

f~ 

007^: 

0084 

oo95 

0106 

011-7 

0127 

56 

i 

620^ 

6214622662376248 

6259 

55 

L 

oi38 

oi49 

oi59 

01700181 

oi9i 

55 

t 

6270 

6281 

6293  63o4 

63i5 

6326 

54 

t 

0202 

02  1C 

022^ 

O234  0245 

0256 

54 

6 

6337 

6348 

6359637i 

6382 

6393 

53 

6 

0266 

0277 

0288 

0298 

o3o9 

O32O 

53 

8 

6471 

64i5  64266437 
6482  6493  65b4 

6^1 

646o 
6526 

52 

5i 

8 

o33o 
0394 

o34i 
o4o5 

o352 
o4i6 

o362 
0426 

o373 
o437 

o384 

5a 
5i 

9 

6538 

6549656o  6571 

6582 

6593 

5o 

9 

o458 

o469 

o48o 

o49o 

o5oi 

O5l2 

5o 

10 

9.946604 

66i5 

6627  6638  6649 

6660 

49 

10 

9.95o522 

o533 

o544 

o554 

o565 

o576 

49 

ii 

6671 

6682 

669367o467i5 

6-726 

48 

ii 

o586 

o597 

0607 

0618 

0629 

o639 

48 

12 

6738 

6749  6760  6771 

6782 

6793 

47 

12 

o65o 

0661 

0671 

0682 

o69c 

0703 

47 

i3 

68o4 

68i5 

6826  6837 

6849 

6860 

46 

i3 

0-714 

0-724 

o735 

o746 

o756 

0767 

46 

i4 

6871 

6882 

689369o4 

69i5 

6926 

45 

i4 

o778 

0-788 

°799 

o8o9 

0820 

o83i 

45 

i5 

6937 

6948 

6959  697o 

6982 

6993 

44 

1  5 

o84i 

o852 

0862 

o873 

0884 

o894 

44 

16 

7004 

7oi5 

702617037 

7o48 

7°59 

43 

16 

o9o5 

o9i5 

O926 

o937 

o947 

o958 

43 

17 

7o7o 

7081 

70927io3 

7114 

7I25 

42 

i7 

o968 

°979 

o99o 

IOOO 

IOII 

1021 

42 

18 

7i36 

7147 

7i587i7o 

7181 

7192 

4i 

18 

1032 

io43 

io53 

1064 

io74 

io85 

4i 

19 

7203 

7214 

7225  7236 

7247 

7258 

4o 

i9 

io96 

1106 

1117 

1127 

u38 

n48 

4o 

20 

9.947269 

7280 

729I 

7302 

73i3 

7324 

39 

20 

9.95n59 

II7O 

1180 

1  191 

I2OI 

1212 

39 

21 

7335 

73467357 

7368 

7379 

739o 

38 

21 

1222 

1233 

1244 

1254 

1265 

1275 

38 

22 

74oi 

74l2 

7423 

7434 

7445 

7456 

37 

22 

1286 

I296 

i3o7 

.3i7 

i328 

i339 

3? 

23 

7467 

7478;7489 

75oo 

75n 

7522 

36 

23 

i349 

i36o 

i37o 

i38i 

i39i 

1402 

36 

24 

7533 

7545 

7556 

7567 

7578 

7589 

35 

e4 

1412 

i423 

i434 

1  444 

i455 

i465 

35 

25 

76oo 

76n 

-7622 

7633 

7644 

7655 

34 

25 

i476 

i486 

1497 

1507 

i5i8 

i528 

34 

26 

7665 

7676 

7687 

7698 

77o9 

7720 

33 

26 

i539 

i549 

:56o 

i57o 

i58i 

i59i 

33 

27 

773i 

7742 

7753 

7764 

7775 

7786 

32 

27 

1602 

i6i3 

1623 

1  634 

1  644 

i655 

32 

28 

7797 

-7808 

78i9 

783o 

7852 

3i 

28 

i665 

1676 

1686 

i697 

I7o7 

1718 

3i 

29 

7863 

7874 

7885 

7896 

7907 

79l8 

3o 

29 

1728 

i739 

i749 

1760 

I77o 

1781 

3o 

3o 

9.947929 

7940 

795i 

7962 

7973 

7984 

29 

3o 

9.95i79i 

1802 

1812 

1823 

i833 

1  844 

29 

3i 

7995 

8006  8017  8028 

8o38 

8049 

28 

3i 

1  854 

i865 

1875 

1886 

i896 

J9°7 

28 

32 

8060 

8071 

8082  8o93 

8io4 

8n5 

27 

32 

1917 

I928 

i938 

i949 

i959 

i969 

27 

33 

8126 

81378148 

8i59 

8170 

8181 

26 

33 

i98o 

i99o 

20OI 

2OII 

2022 

2032 

26 

34 

8l92 

82o382i3 

8224 

8235 

8246 

25 

34 

2o43 

2o53 

2064 

2074 

2085 

2O95 

25 

35 

8257 

8268  8279 

829o 

83oi 

83i2 

24 

35 

2106 

2116 

2126 

2l37 

2l47 

2i58 

24 

36 

8323 

8334  8344 

8355 

8366 

8377 

23 

36 

2168 

2179 

2i89 

22OO 

2210 

2221 

23 

37 

8388 

83998410 

8421 

8432 

8443 

22 

37 

223l 

2252 

2262 

2273 

2283 

22 

38 

8454 

8464 

8475 

8486 

8497 

85o8 

21 

38 

2294 

23o4 

23i4 

2325 

2335 

2346 

21 

39 

85i9 

853o854i 

8552 

8562 

8573 

2O 

39 

2356 

2367 

2377 

2387 

2398 

2408 

20 

4o 

9.  948584 

8595  8606 

8617 

8628 

863o 

I9 

4o 

9.9524i9 

2440 

245o 

2460 

2471 

X9 

4i 

865o 

86608671 

8682 

8693  8704 

18 

4i 

248  1 

2492 

25O2 

25l2 

2523 

2533 

18 

42 

87i5 

87268736 

8747 

8758  8769 

'7 

42 

2544 

2554 

2565 

2575 

2585 

2596 

17 

43 

8780 

8791 

8802 

8812 

8823  8834 

16 

43 

2606 

2617 

2627 

2637 

2648 

2658 

16 

44 

8845 

88568867 

8878 

8888 

8899 

i5 

44 

2669 

2679 

2689 

2700 

27IO 

2720 

i5 

45 

8910 

8921 

8932 

8943 

8954 

8964 

i4 

45 

2731 

274l 

2752 

2762 

2772 

2783 

i4 

46 

8975 

8986  Sogri 

9oo8 

9oi9 

9029 

i3 

46 

2793 

2803 

2814 

2824 

2835 

2845 

i3 

47 

9040 

9o5i 

9o62 

9o73 

9o839o94 

12 

47 

2855 

2866 

2876 

2886 

2897 

2907 

12 

48 

9io5 

9116 

9I27 

9i38 

9i489i59 

II 

48 

2918 

2928 

2938 

2949 

2959 

2969 

II 

49 

9I7° 

9181 

9I92 

92O2 

92l3 

9224 

IO 

49 

298o 

299° 

3ooo 

3on 

3O2I 

3o3i 

IO 

5o 

9.949235 

9246 

9256 

9267 

9278 

9289 

9 

5o 

9>953o42 

3o52 

3o62 

3o73 

3o83 

3o93 

9 

5i 

93oo 

93io 

932I 

9332 

9343 

D354 

8 

5i 

3io4 

3n4 

3i24 

3i35 

3i45 

3i55 

8 

52 

9364 

9375 

9386 

9397 

94o894i8 

7 

52 

3i66 

3i76 

3i86 

3i97 

3207 

3217 

7 

53 

9429 

944o 

945i 

9472  9483 

6 

53 

3228 

3238 

3248 

3259 

3269 

3279 

6 

54 

9494 

95o5 

95i5 

9526 

9537  9548 

5 

54 

329o 

33oo 

33io 

332i 

333i 

334i 

5 

55 

9558 

9569 

958o 

959i 

96o2  9612 

4 

55 

3352 

3362 

3372 

3382 

3393 

34o3 

4 

56 

9623 

9634 

9645 

9655 

9666  96-7-7 

3 

56 

34i3 

3424 

3434 

3444 

3455  3465 

3 

5/ 
58 

9688 
9752 

9698 
9763 

97099720 

9774  9784 

973i'974i 
9-795  9806 

2 
I 

57 

58 

34?5 
3537 

3485  3496  35o6J35i6  3527 
35473557  3568  3578  3588 

2 

I 

59 

9816982-7 

98">8  9849 

985998-70 

O 

59 

3599  36o9  36i9  3629'364o  365o  o 

60"     50" 

40"   30"   20"   10" 

cf 

60"     50"   40"   30"   20"   10"   a. 

Co-sine  of  27  Degrees. 

Co-sine  of  26  Degrees. 

PP^t*1"  2"  3//  4"  5"  6"  ?"  8"  9" 

.  (  I"  2"  3"  4"  5"  6"  7"  8"  9" 

fft{  1   2   3 

4   5   7   8   9  10      irl  £12   3   456   789 

LOGARITHMIC    T  v  \  G  E  N  r  s. 


87 


jj 

Tangent  of  62  Degrees. 

.5 

Tangent  of  03  Degrees. 

s 

0"     |  10" 

20" 

30" 

40" 

50" 

s 

0" 

10" 

20" 

30" 

40" 

50" 

o 

10.274326 

4376 

442  7 

4478 

4529 

458o 

59 

o 

10.292834 

2886 

2938 

2990 

3o42 

3o94 

59 

I 

463o 

468i 

4732 

4783 

4834 

4885 

58 

i 

3i46 

3i99 

325i 

33o3 

3355 

3407 

58 

2 

4935 

4986 

5o37 

5o88 

5i39 

Sigo 

57 

2 

3459 

35ii 

3563 

36i5 

3667 

3720 

57 

3 

6240 

5291 

5342 

SSgS 

5444 

5495 

56 

3 

3772 

3824 

3876 

392& 

398o 

4o32 

56 

4 

5546 

5597 

5647 

5698 

5749 

58oo 

55 

4 

4o84 

4i37 

4189 

424i 

4293 

4345 

55 

5 

585i 

5902 

5953 

6oo4 

6o55 

6io5 

54 

5 

4397 

4449 

45o2 

4554 

46o6 

4658 

54 

6 

6i56 

62O7 

6258 

6309 

636o 

64n 

53 

6 

4710 

4763 

48i5 

4867 

49i9 

4971 

53 

7 

6462 

65i3 

6564 

66i5 

6666 

6717 

52 

7 

5o24 

5o76 

5i28 

5i8o 

5232 

5285 

5a 

8 

6768 

6819 

6870 

6920 

697i 

7O22 

5i 

8 

5337 

5389 

544  1 

5494 

5546 

5598 

5.i 

9 

7o73 

7I24 

7i75 

7226 

7277 

7328 

5o 

9 

565o 

57o3 

5755 

58o7 

5859 

5912 

5o 

10 

io.277379 

743o 

748  1 

7532 

7583 

7634 

4c; 

10 

10.295964 

6016 

6068 

6121 

6i73 

6225 

49 

ii 

7685 

7736 

7787 

7838 

7889 

794o 

48 

1  1 

6278 

633o 

6382 

6434 

6487 

6539 

48 

12 

799  r 

8o43 

8094 

8i45 

8i96 

8247 

47 

12 

659i 

6644 

6696 

6748 

6801 

6853 

47 

13 

8298 

8349 

84oo 

845  1 

85o2 

8553 

46 

i3 

6905 

6958 

7OIO 

7o62 

7ii5 

7167 

46 

i4 

86o4 

8655 

8706 

8757 

88o9 

8860 

45 

i4 

7219 

7272 

7324 

7377 

7429 

748i 

45 

i5 

8911 

8962 

9013 

9064 

9u5 

9166 

44 

i5 

7534 

7586 

7638 

769i 

7743 

7796 

44 

16 

9217 

9268 

9320 

937i 

9422 

9473 

43 

16 

•  7848 

79oo 

7953 

8oo5 

8o58 

8no 

43 

i? 

9524 

9575 

9626 

9678 

9729 

9780 

42 

J7 

8i63 

82i5 

8267 

8320 

8372 

8425 

42 

18 

9831 

9882 

9933 

9984 

..36 

..87 

4i 

18 

8477 

853o 

8582 

8635 

8687 

874o 

4i 

*9 

io.28oi38 

0189 

0240 

0292 

o343 

o394 

4o 

19 

8792 

8845 

8897 

8949 

9OO2 

9o54 

4o 

2t_/ 

io.28o445 

0496 

o548 

o599 

o65o 

0701 

39 

20 

IO.299I07 

9i59 

92I2 

9264 

93i7 

937o 

39 

21 

0752 

o8o4 

o855 

0906 

o957 

1009 

38 

21 

9422 

9475 

9527 

958o 

9632 

9685 

38 

22 

1060 

mi 

1162 

I2l4 

1265 

i3i6 

37 

22 

9737 

979° 

9842 

9895 

9947 

.... 

37 

23 

i367 

1419 

i47o 

l52I 

l572 

1624 

36 

23 

io.3ooo53 

oio5 

oi58 

O2IO 

0263 

o3i5 

36 

24 

i675 

1726 

i777 

1829 

1880 

1931 

35 

24 

o368 

0421 

o473 

o526 

o578 

o63i 

35 

25 

1983 

2034 

2o85 

2I37 

2188 

2239 

34 

25 

o684 

o736 

o789 

o84i 

o894 

o947 

34 

26 

2291 

2342 

2393 

2445 

2496 

2547 

33 

26 

°999 

1052 

iio5 

n57 

1210 

1263 

33 

27 

2599 

265o 

2701 

2753 

2804 

2855 

32 

27 

i3i5 

i368 

1421 

i473 

i526 

i579 

32 

28 

2907 

2958 

3oo9 

3o6i 

3lI2 

3i64 

3i 

28 

i63i 

i684 

i737 

1789 

1842 

i895 

3i 

29 

32i5 

3266 

33i8 

3369 

3421 

3472 

3o 

29 

i947 

2OOO 

2o53 

2106 

2i58 

2211 

3o 

3o 

ic.»83523 

3575 

3626 

3678 

3729 

378o 

29 

3o 

IO.  3O2264 

23i6 

2369 

2422 

2475 

2527 

29 

3i 

3832 

3883 

3935 

3986 

4o38 

4089 

28 

3  1 

258o 

2633 

2686 

2738 

279I 

2844 

28 

32 

4i4o 

4192 

4243 

4295 

4346 

4398 

27 

32 

2897 

295o 

3OO2 

3o55 

3io8 

3i6i 

27 

33 

4449 

45oi 

4552 

46o4 

4655 

47o7 

26 

33 

32i3 

3266 

33i9 

3372 

3425 

3478 

26 

34 

475S 

48io 

486i 

49>3 

4964 

5oi6 

25 

34 

353o 

3583 

3636 

3689 

3742 

3794 

25 

35 

5067 

5119 

5170 

5222 

5273 

5325 

24 

35 

3847 

39oo 

3953 

4oo6 

4o59 

4lI2 

24 

36 

5376 

5428 

5479 

553i 

5582 

5634 

23 

36 

4i64 

42  I  7 

4270 

4323 

4376 

4429 

23 

37 

5686 

5737 

5789 

584o 

5892 

5943 

22 

37 

4482 

4535 

4588 

464o 

4693 

4746 

22 

38 

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6o46 

6098 

6i5o 

6201 

6253 

21 

38 

4799 

4852 

49o5 

4958 

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5o64 

21 

39 

63o4 

6356 

64o8 

6459 

65ii 

6562 

20 

39 

5i  i7 

5170 

5223 

5276 

5328 

538i 

20 

4o 

10.286614 

6666 

6717 

6769 

6821 

6872 

19 

4o 

io.3o5434 

5487 

554o 

5593 

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19 

4i 

6924 

6975 

7027 

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7i3o 

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18 

4i 

5752 

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5858 

59ii 

5964 

6oi7 

18 

42 

7234 

7285 

7337 

7389 

744o 

7492 

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42 

6o7o 

6128 

6i76 

6229 

6282 

6335 

17 

43 

7544 

7595 

7647 

7699 

775i 

7802 

16 

43 

6388 

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6  49  4 

6547 

6600 

6654 

16 

44 

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79o6 

7957 

8009 

8061 

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44 

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6760 

68i3 

6866 

6919 

6972 

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4? 

8i64 

8216 

8268 

8319 

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8423 

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45 

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7078 

7i3i 

7i84 

7237 

729o 

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46 

8475 

8526 

8578 

863o 

8682 

8733 

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46 

7344 

7397 

745o 

75o3 

7556 

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47 

8785 

8837 

8889 

8941 

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9044 

12 

47 

7662 

77i5 

7768 

7822 

7875 

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12 

48 

9096 

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925i 

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48 

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49 

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10 

49 

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10.290029 

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7 

52 

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9364 

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7 

53 

o65r 

o7o3 

0755 

o8o7 

oSS^ 

0911 

6 

53 

9577 

963i 

9684 

9737 

979° 

9844 

6 

54 

0963 

ioi5 

1066 

1118 

II70 

1222 

5 

54 

9897 

995o 

...4 

..57 

.  no 

.i64 

5 

55 

I274 

i326 

i378 

i43o 

1482 

i534 

4 

55 

IO.3lO2I7 

027O 

0324 

o377 

o43o 

o484 

4 

56 

i586 

i638 

1690 

I742 

1-794 

i846 

3 

56 

o537 

oSgO 

o644 

o697 

o75o 

0804 

3 

5? 

1898 

i95o 

2002 

2o54 

2106 

2i58 

2 

57 

o857 

0910 

o964 

IOI7 

io7o 

1124 

2 

58 

22IO 

2262 

23l4 

2366 

2418 

2470 

I 

58 

ii77 

I23l 

1284 

i337 

i39i 

i444 

I 

59 

2522 

2574 

2626 

2678 

273o 

2782 

O 

59 

i498 

i55i 

i6o5 

i658 

1711 

I765 

0 

60" 

50"   40" 

30" 

20" 

10" 

a 

«0"      50" 

40" 

30" 

20" 

10" 

a 

Co-tangent  of  27  Degrees. 

.9 

s 

Co-tangent  of  26  Degrees. 

P  P  ,rt  5  l"   ~"   3"   4//   5//   6"   7//   8"  9"   1        *  *  1"   2"   3"   4//   r>"   (J"   7"   8"  9" 

tri)  5   10  15  21  26  31  36  41  46  |  l  '  l  urt  {  5   11  16  21  <M  32  37  43  47 

88 


LOGARITHMIC    SINES. 


d      Sine  of  64  Degrees.      , 

.3 

Sine  of  G5  Degrees. 

a   0" 

10" 

20" 

30" 

40*' 

50"  1 

* 

0" 

10"   20"  1  30" 

40" 

50" 

o  9.953660 

367o 

368i 

369i 

37oi 

3712 

59 

o 

9.  957276 

7286 

7295 

73o5 

73i5 

7325 

59 

I 

3722 

3732 

3742 

3753 

3763 

3773 

58 

i 

7335 

7344 

7354 

7364 

7374 

7384 

58 

2 

3783  3794 

38o4 

38i4 

3824 

3835 

57 

2 

7393 

74o3 

74i3 

7423 

7433 

7442 

57 

3 

3845  3855 

3865 

3876 

3886 

3896 

56 

3 

7452 

7462 

7472 

7482 

7491 

75oi 

56 

4 

3906  39i7 

3927 

3937 

3947 

3957 

55 

4 

75n 

752I 

753i 

754o 

755o 

756o 

55 

5 

3968 

3978 

3988 

3998 

4009  4019 

54 

5 

757o 

7579 

7589 

7599 

7609 

76l9 

54 

6 

4029 

4o39 

4o5o  4o6o 

4o7o  4o8o 

53 

6 

7628 

7638 

7648 

7658 

7667 

7677 

53 

7 

4090 

4ioi 

4m  4121 

4i3i 

4i4i 

52 

7 

7687 

7697 

77°7 

77i6 

7726 

7736 

52 

8 

4i52 

4162 

41-724182 

4192 

4203 

5i 

8 

7746 

7755 

7765 

7775 

7785 

7794 

5i 

9 

42i3 

4223 

42334243 

4254 

4264 

5o 

9 

78o4 

7814 

7824 

7833 

7843 

7853 

5o 

10 

9.9542-74 

4284 

42944305 

43i5 

4325 

49 

10 

9.95-7863 

7872 

-7882 

7892 

7902 

7911 

49 

ii 

4335 

4345 

4356 

4366 

4376 

4386 

48 

ii 

792I 

793i 

794o 

795o 

7960 

797° 

48 

12 

4396 

44o6 

44i7 

4427 

4437 

4447 

47 

12 

7979 

7989 

7999 

8oo9 

8018 

8028 

47 

i3 

4457 

4468 

44  7  8 

4488 

4498 

45o8 

46 

i3 

8o38 

8047 

8o57 

8o67 

8077 

8086 

46 

i4 

45i8 

4529 

4539 

4549 

4559 

4569 

45 

U 

8o96 

8106 

8n5 

8i25 

8i35 

8i45 

45 

:5 

4579 

4589 

46oo!46io 

4620 

463o 

44 

i5 

8i54 

8i64 

8i74 

8i83 

8i93 

8203 

44 

16 

464o 

465o 

466i 

467i 

468i 

469i 

43 

16 

8213 

8222 

8232 

8242 

825i 

8261 

43 

17 

47oi 

47n 

472I 

4732 

4742 

4752 

42 

17 

8271 

8280 

829,0 

83oo 

83o9 

83i9 

42 

18 

4762 

4772 

4782 

4-792 

4802 

48i3 

4i 

18 

8329 

8339 

8348 

8358 

8368 

8377 

4i 

19 

4823 

4833 

4843 

4853 

4863 

4873 

4o 

19 

8387 

8397 

84o6 

84i6 

8426 

8435 

4o 

20 

9.954888 

4894 

4904 

4914 

4924 

4934 

39 

20 

9.958445 

8455 

8464 

8474 

8484 

8493 

39 

21 

4944 

4954 

4964 

4974 

4985 

4995 

38 

21 

85o3 

85i3 

8522 

8532 

8542 

855i 

38 

22 

5oo5 

5oi5 

5o25 

5o35 

5o45 

5o55 

37 

22 

856i 

857i 

858o 

859o 

8600 

8609 

37 

23 

5o65 

5o75 

5o86 

5o96 

5io6 

5n6 

36 

23 

86i9 

8628 

8638 

8648 

8657 

8667 

36 

24 

5i26 

5i36 

5i465i56 

5i66 

5i76 

35 

24 

86778686 

8696 

8-706 

87I5 

8725  35 

25 

5i86 

5i96 

5207  52I7 

5227 

5237 

34 

25 

8734 

8744 

8754 

8763 

8773 

8783 

34 

26 

5247 

5257 

5267 

5277 

5287 

5297 

33 

26 

8792 

8802 

8812 

8821 

883i 

884o 

33 

27 

53o7 

53  1  7 

5327 

5337 

5348 

5358 

32 

27 

885o 

8860 

8869 

8879 

8888 

8898 

32 

28 

5368 

5378 

5388 

5398 

54o8 

54i8 

3i 

n 
2O 

89o8 

8917 

8927 

8937 

8946 

8956 

3i 

29 

5428 

5438 

5448 

5458 

5468 

5478 

3o 

29 

8965 

8975 

8985 

8994 

9oo4 

9013 

3o 

3o 

9.955488 

5498 

5  5  08 

55i8 

5528 

5538 

29 

3o 

9.959O23 

9o33 

9042 

9052 

9o6i 

9o7i 

29 

3i 

5548 

5559 

5569 

5579 

5589 

5599 

28 

3i 

9080 

9090 

9100 

9109 

9119 

9128 

28 

32 

5609 

56i9 

5629  5639 

5649 

5659 

27 

32 

9i38 

9148 

9i57 

9167 

9i76 

9186 

27 

33 

5669 

5679 

5689 

5699 

57o9 

57i9 

26 

33 

9i95 

9205 

92l5 

9224 

9234 

9243 

26 

34 

5729 

5739 

5749 

5759 

5769 

5779 

25 

34 

9253 

9262 

9272 

9282 

9291 

93oi 

25 

35 

5789 

5799 

58095819 

5829 

5839 

24 

35 

93io 

9320 

9329 

9339 

9348 

9358 

24 

36 

5849 

5859 

5869 

5879 

5889 

5899 

23 

36 

9368 

9377 

9387 

9396 

9406 

94i5 

23 

37 

59o9 

59i9 

5929 

5939 

5949 

5959 

22 

37 

9425 

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9453 

9463 

9473 

22 

38 

5969 

5979 

5989 

5999 

6009 

6oi9 

21 

38 

9482 

9492 

95oi 

95ii 

9520 

953o 

21 

39 

6o29  6o39 

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6069 

6o79 

2O 

39 

9539 

9549 

9558 

9568 

9577 

9587 

2O 

4o 

4i 

9.  956089  6099 

6i486i58 

6108.6118 
6168  6178 

6128 
6188 

6i38 
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Is 

4o 
4i 

9.959596 
9654 

9606 
9663 

96i5 
9673 

9625 
9682 

9634 
9692 

9644 
97oi 

'9 

18 

42 

62086218 

6228:6238 

6248 

6258 

17 

42 

9711 

9720 

973o 

9739 

9749 

9758 

17 

43 

6268 

62?8 

6288 

6298 

63o8 

6317 

16 

43 

9768 

9777 

9787 

9796 

9806 

9815 

16 

44 

6327 

6337 

6347 

6367 

6377 

i5 

44 

9825 

9834 

9844 

9853 

9863 

9872 

i5 

45 

6387 

6397 

64o7 

64i7 

6427 

6437 

i4 

45 

9882 

989i 

99oo 

9910 

9919 

9929 

i4 

46 

6447 

6457 

6466 

6476  6486 

6496 

i3 

46 

9938 

9948 

9957 

9967 

9976 

9986 

i3 

47 

65o6 

65i6 

6526 

6536 

6546 

6556 

12 

47 

9995 

...5 

..24 

..33 

..43 

12 

48 

6566 

6575 

6585 

65g5 

66o5 

66i5 

II 

48 

9.960052 

0061 

Of  7I 

0080 

0090 

0099 

II 

49 

6625 

6635 

6645 

6655  6665 

6674 

IO 

49 

0109 

0118 

0128 

oi37 

oi47 

oi56 

IO 

5o 

9.956684 

6694 

67o4 

67146724 

6734 

9 

5o 

9.960165 

0175 

0184 

0194 

0203 

02l3 

9 

5i 

6744 

6754 

6763 

6773 

6783 

6793 

8 

5i 

O222 

0232 

0241 

O25o 

0260 

0269 

8 

52 

68o3 

68i3 

6823 

6833 

6843 

6852 

7 

52 

0279 

0288 

0298 

0307  o3i7 

o326 

7 

53 

6862 

6872 

6882 

6892 

6902 

6912 

6 

53 

o335 

o345 

o354 

o364o373 

o382 

6 

54 

6921 

693i 

6941 

695i 

6961 

697i 

5 

54 

o3g2 

o4oi 

o4n 

0420  o43o 

o439 

5 

55 

6981 

6990 

7ooo 

7010 

7020 

7o3o 

4 

55 

o448  o458 

0467 

0477  o486  o495 

4 

56 

7o4o 

7o5o 

7o59 

7069 

7079 

7080 

3 

56 

o5o5Jo5i4 

o524 

o533o542  o552 

3 

57 

7°99 

-7109 

7118 

7128 

7i38 

7i48 

2 

57 

o56i 

0571 

o58o 

oSSg  0599  0608 

2 

58 

7i58 

7i68 

7i77 

7187 

7197 

7207 

I 

58i    0618 

0627 

o636 

o646  o655  o665 

I 

5o 

7217 

7227 

7236 

7246  7256 

7266 

0 

59 

oG74 

o683 

o693 

0702  0711  0721 

0 

60' 

50" 

40" 

30"   20"   10" 

s 

60"     50"  |  40" 

30"  !  20"   10"   ^ 

Co-sine  of  25 

Degrees. 

Co-sine  of  24  Degrees.     a 

f  ]"  o"  3"  4" 

5"  6"  7"  8"  9" 

PPartJ  l"  ~"  3//  4"  5"  G"  7"  8"  •'" 

P.Fart^  i  g'  3  4 

56789 

irl\  1   2   3   4   5   6   7   S   <> 

LOGARITHMIC    TANGEJVTS. 


89 


J 

Tangent  of  64  Degrees. 

a 

Tangent  of  03  Degrees. 

s 

0" 

10"   20" 

30" 

40" 

50" 

& 

0" 

10" 

20" 

30" 

40" 

50" 

~~o|io.3ii8i8 

1872 

1925 

I979 

2O32 

2085 

59 

O 

io.33i327 

i382 

i437 

1492 

1  547 

1602 

59 

I     2i39 

2192 

2246 

2299 

2353 

2406 

58 

I 

i657 

1712 

1767 

1822 

i877 

1932 

58 

21       246O 

25i3 

2567 

2620 

2674 

2727 

57 

2 

i987 

2042 

2097 

2I53 

2208 

2263 

57 

3     2781 

2334 

2888 

2941 

2995 

3o48 

56 

3     23i8 

2373 

2428 

2483 

a538 

2593 

56 

L     3io2 

3i55 

3209 

3263 

33i6 

337o 

55 

4 

2648 

2703 

2758 

28i3 

2868 

2924 

55 

5,     3423 

3477 

353o 

3584 

3637 

369i 

54 

5 

2979 

3o34 

3089 

3i44 

3i99 

3254 

54 

e 

374^ 

3798 

3852 

39o5 

3959 

4oi3 

53 

6 

3309 

3364 

3420 

3475 

353o 

3585 

53 

7 

4o66 

4l2O 

4173 

4227 

4281 

4334 

52 

7 

364o 

3695 

375i 

38o6 

386i 

3916 

52 

8 

4388 

4442 

4495 

4549 

46o3 

4656 

5i 

8 

3971 

4026 

4082 

4i37 

4192 

4247 

5i 

9 

4710 

4764 

48i7 

487i 

4925 

4978 

5o 

9 

4302 

4358 

44i3 

4468 

4523 

45-79 

5o 

10 

io.3i5o32 

5o86 

5  1  39 

SigS 

5247 

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49 

10 

io.334634 

4689 

4744 

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4910 

49 

ii 

5354 

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546i 

55i5 

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48 

1  1 

4g65 

5021 

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48 

12 

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5730 

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12 

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13 

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46 

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5795 

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632i 

6375 

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45 

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6072 

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45 

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6644 

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6806 

6860 

6913 

44 

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6293 

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44 

16 

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43 

16 

6625 

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43 

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7290 

7344 

7398 

7452 

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42 

17 

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18 

76i3 

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10.318260 

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39 

20 

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21 

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21 

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38 

22 

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37 

22 

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37 

23 

9232 

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36 

23 

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36 

24 

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25 

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34 

25 

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8 

52 

8694 

8749 

88o3 

8858 

89i3 

8968 

7 

52 

8703 

8759 

88i5 

8872 

8928 

8985 

7 

53 

9023 

9077 

9132 

9l87 

9242 

9297 

6 

53 

9o4i 

9098 

9i54 

9211 

9267  9324 

6 

54 

935  1 

9406 

946i 

95i6 

957i 

9625 

5 

54 

9380 

9437 

9493 

955o 

9606 

91363 

5 

55 

9680 

9735 

979° 

9845 

99oo 

9955 

4 

55 

9719 

9776 

9832 

9889 

9945 

.  .  .2 

4 

56 

10.330009 

006/4 

on9 

oi74 

0229 

0284 

3 

56 

io.35oo58 

on5 

0171 

O2a8 

0285 

o34i 

3 

57 

o339 

oSg/j 

o448 

o5o3 

o558 

o6i3 

2 

^7 

o398 

o454 

o5n 

o567 

0624 

0681 

2 

58 

0668 

0723 

0778 

o833 

0888 

o943 

I 

58 

o737 

0794 

o85o 

090-7 

0964 

I02O 

I 

69 

0998 

io53 

1  108 

n63 

1218 

1272 

o 

59 

1077 

1  1  34 

1190 

1247 

i3o4 

i36o 

O 

60" 

50" 

40" 

30" 

20"   10" 

S* 

60" 

50" 

40"   30" 

20" 

10" 

R 

Co-tangent  of  25  Degrees. 

s 

Co-tangent  of  24  Degrees. 

g 

p  p   41"  2"  3"  4"  5"  6"  7"  8'  9" 
1  \  5  11  13  <i2  27  33  38  43  49 

C  1"  2"  3"  4"  5"  6"  7"  8"  .0" 
P.  I  art  j  6  n  17  oo  28  33  39  45  50 

90 


L.OGARI1HMIC 


Fj 

Sine  ot  66  Degrees. 

_c 

Sine  of  67  Degrees. 

u? 

0"    |  10"  |  20" 

30" 

40"   50" 

2 

0" 

10"  |  20"  |  30" 

40" 

50" 

~0 

9.960730 

0740 

0749 

o758 

o768  o777 

59 

o  9.964026 

4o35 

4o44 

4o53 

4o62 

4o7i 

59 

I 

0786 

0796 

o8o5 

0814 

0824 

o833 

58 

I 

4o8o 

4089 

4o98 

4io6 

4u5 

4124 

58 

2 

0843 

o852 

0861 

o87i 

0880 

o889 

57 

2 

4i33 

4i42 

4i5i 

4i6o 

4169 

4i78 

57 

3 

0899 

0908 

0917 

092-7 

0936 

o945 

56 

3 

4i87 

4196 

42o5 

4214 

4222 

423l 

56 

4 

O955 

0964 

o973 

0983 

0992 

1002 

55 

4 

4240 

4249 

4258 

4267 

4276 

4285 

55 

5!    ion 

IO2O 

io3o 

1039 

io48 

io58 

54 

5 

4294 

43o3 

43n 

4320 

4329 

4338 

54 

6 

1067 

1076 

1086 

1095 

1104 

iii3 

53 

6 

4347 

4356 

4365 

4374 

4383 

4392 

53 

7 

1123 

Il32 

n4i 

n5i 

1160 

u69 

52 

7 

44oo 

44o9 

44i9 

4427 

4436 

4445 

52 

8 

1179 

1188 

1197 

1207 

1216 

1225 

5i 

8 

4454 

4463 

447i 

448o 

4489 

4498 

5i 

9 

1235 

1244 

1253 

1263 

I272 

I28l 

5o 

9 

4507 

45i6 

4525 

4534 

4542 

455i 

5o 

:o 

9.961290 

i3oo 

i3og 

i3i8 

1328 

i337 

49 

IO 

9.  964560 

4569 

4578 

4587 

4596 

46o4 

49 

ii 

1  346 

i356 

i365 

i374 

i383 

i393 

48 

ii 

46i3 

4622 

463i 

464o 

4649 

4658 

48 

12 

1402 

i4n 

1421 

i43o 

i439 

i448 

47 

12 

4666 

4675 

4684 

4693 

47o? 

47n 

47 

i3 

i458 

1467 

1476 

i485 

i495 

i5o4 

46 

i3 

4720 

4728 

4737 

4746 

4755 

4764 

46 

i4 

i5i3 

i523 

i532 

i54i 

i55o 

i56o 

45 

i4 

4773 

478i 

479o 

4799 

48o8 

48i7 

45 

i5 

1569 

i578 

1587  1597 

1606 

i6i5 

44 

i5 

4826 

4834 

4843 

4852 

486i 

487o 

44 

16 

1624 

1  634 

i643  i652 

1661 

i67i 

43 

16 

4879 

4887 

4896 

49o5 

49i4 

4923 

43 

r? 

1680 

1689 

i698'i7o8 

i7i7 

I726 

42 

ll 

493i 

494o 

4949 

4958 

4967 

4975 

42 

18 

i735 

1745 

17541763 

I772 

I782 

4i 

18 

4984 

4993 

5OO2 

Son 

5O20 

5028 

4i 

19 

1791 

1800 

1809  1819 

1828 

i837 

4o 

J9 

5o37 

5o46 

5o55 

5o64 

5o72 

5o8i 

4o 

20 

9.961846 

i856 

1865:1874 

i883 

l892 

39 

20 

9.965090 

5o99 

5io7 

5n6 

5is5 

5i34 

39 

21 

1902 

1911 

1920:1929 

I939 

i948 

38 

21 

5i43 

5i5i 

5i6o 

5i69 

5i78 

5i87 

38 

22 

i957 

1966 

I975ji985 

i994 

2OO3 

37 

22 

SigS 

5204 

52i3 

5222 

523o 

5239 

37 

23 

2OI2 

2O2I 

2o3l  2O40 

2049 

2o58 

36 

23 

5248 

5257 

5266 

5274 

5283 

5292 

36 

3/ 

2067 

2077 

2O86  2O95 

2104 

2Il3 

35 

24 

53oi 

53o9 

53i8 

5327 

5336 

5344 

35 

25 

2123 

2132 

2l4l  2l5o 

2i59 

2l69 

34 

25 

5353 

5362 

537i 

5379 

5388 

5397 

34 

26 

2178 

2187 

2196  22O5 

22l4 

2224 

33 

26 

54o6 

54i4 

5423 

5432 

544  1 

5449 

33 

27 

2233 

2242 

225l  226O 

2269 

2279 

32 

27 

5458 

5467 

5476 

5484 

5493 

55o2 

32 

28 

2288 

2297 

23o6  23i5 

2325 

2334 

3i 

28 

55ii 

55i9 

5528 

5537 

5546 

5554 

3  1 

29 

2343 

2352 

236i  237o 

2379 

2389 

3o 

29 

5563 

5572 

558o 

5589 

5598 

56o7 

3o 

3c 

9.962398 

2407 

2416  2425 

2434 

2444 

29 

3o 

9.9656i5 

5624 

5633 

5642 

565o 

5659 

29 

3i 

2453 

2462 

2471  2480 

2489 

2498 

28 

3i 

5668 

5676 

5685 

5694 

5702 

57n 

28 

32 

25o8 

2517 

25262535 

2544 

2553 

27 

32 

5720 

5729 

5737 

5746 

5755 

5763 

27 

33 

2562 

2572 

258i  2590 

2599 

2608 

26 

33 

5772 

578i 

579o 

5798 

58o7 

58:6 

26 

34 

2617 

2626 

2635  2645 

2654 

2663 

25 

34 

5824 

5833 

5842 

585o 

5859 

5868 

25 

35 

2672 

2681 

2690  2699 

27o8 

2717 

24 

35 

5876 

5885 

5894 

59O2 

59n 

5920 

24 

36 

2727 

2736 

2745  2754 

2763 

2772 

23 

36 

5929 

5937 

5946 

5955 

5963 

5972 

23 

37 

2781 

2790 

2799  2809 

2818 

2827 

22 

37 

598i 

5989 

5998 

6oo7 

6oi5 

6024 

22 

38 

2836 

2845 

28542863 

2872 

2881 

21 

38 

6o33 

6o4i 

6o5o 

6o59 

6o67 

6o76 

21 

39 

2890 

2899 

2909  2918 

2927 

2936 

2O 

39 

6o85 

6o93 

6102 

6m 

6n9 

6128 

20 

4o 

9.962945 

2954 

2963  2972 

298i 

299O 

19 

4o 

9.966i36 

6i45 

6i54 

6162 

6i7i 

6180 

1  9 

4i 

2999 

3oo8 

3oi8;3o27 

3o36 

3o45 

18 

4i 

6188 

6i97 

6206 

6214 

6223 

6232 

18 

42 

3o54 

3o63 

3o72  3o8i 

3o9o 

3o99 

17 

42 

6240 

6249 

6257 

6266 

6275 

6283 

J7 

43 

3io8 

3ii7 

3i26:3i35  3i44 

3i53 

16 

43 

6292 

63oi 

63o9 

63i8 

6326 

6335 

16 

44 

3i63 

3172 

3i8i  3190  3199 

3208 

i5 

44 

6344 

6352 

636i 

637o 

6378 

6387 

i5 

45 

3217 

3226 

323532443253 

3262 

i4 

45 

6395 

64o4 

64i3 

6421 

643o 

6438 

i4 

46 

3271 

3280 

3289:3298 

33o7 

33i6 

i3 

46 

6447 

6456 

6464 

6473 

6482 

649o 

i3 

4? 

3325 

3334 

3343  3352 

336i 

3370 

12 

47 

6499 

65o7 

65i6 

6525 

6533 

6542 

12 

48 

3379 

3388 

3398 

34o7 

34i6 

3425 

II 

48 

655o 

6559 

6567 

6576 

6585 

6593 

I  I 

49 

3434 

3443 

3452 

346i 

347o 

3479 

IO 

49 

6602 

6610 

66i9 

6628 

6636 

6645 

10 

5o 

9.963488 

3497 

35o6 

35i5 

3524 

3533 

9 

5o 

9.Q66653 

6662 

667o 

6679 

6688 

6696 

9 

5i 

3542 

355i 

356o 

3569 

3578 

3587 

8 

5i 

67o5 

67i3 

6722 

673o 

6739 

6748 

8 

62 

3596 

36o5 

36i4 

3623 

3632 

364i 

7 

52 

6756 

6765 

G773 

6782 

679o 

6799 

7 

53 

365o 

3659 

3668 

3677 

3686 

3695 

6 

53 

6808 

6816 

6825 

6833684s 

685o 

6 

54 

3704 

37i3 

3722 

373o 

3739 

3748 

5 

54 

6859 

6867 

6876 

68846898 

69O2 

5 

55 

3757 

3766 

3775 

37843793 

38o2 

4 

55 

6910 

69i9 

6927 

6936  6944 

6953 

4 

56 
*7 

38ii 
3865 

3820 

3874 

3829 
3883 

3838.  3847 
3892  3901 

3856 
39io 

3 

2 

56 

5; 

6961 
7oi3 

697o6978 
7o2i  7o3o 

6987  6995 
7o38  7o47 

7oo4 
7o55 

3 

2 

58 

39i9 

3928 

3937 

39463955 

3963 

I 

58 

7o64 

7o72  7o8i 

7o89  7o98 

7106 

I 

59 

3972 

398i 

3990 

3999  4oo8 

4017 

O 

59 

7i  i5 

7I23|7l32 

7i4oi7i49  7i57 

0 

60"     50" 

40" 

30"   20" 

10" 

a 

60"    I  50"   40" 

30"  |  20"   10" 

Co-sine  of  23  Degrees. 

X 

Co-sine  of  22  Degrees. 

Z 

c  ]//  o"  3"  4"  5"  fi"  7"  g"  9" 
l'l**?\  123455678 

.<  1"  2"  3"  4"  5"  6"  7"  S"  9" 
1  )  1   2   3   3   4   5   fi   7   8 

LOGARITHMIC    TANGENTS. 


01 


r 

Tangent  of  CO  Degrees. 

d 

Tangent  of  67  Degrees. 

! 

3 

0" 

10" 

20" 

30" 

40" 

50" 

2 

0" 

10" 

207 

30" 

40- 

50" 

o 

io.35i4i7 

1474 

i53o 

i587 

1  644 

I7OO 

59 

0 

io.372i48 

22O7 

2265 

2324 

2382 

244  1 

59 

I 

1757 

1814 

1870 

1927 

1984 

2O40 

58 

I 

2499 

2558 

2617 

2675 

2734 

2792 

58 

1 

2097 

2i54 

2211 

2267 

2324 

238i 

57 

2 

285i 

29IO 

2968 

3027 

3o85 

3i44 

57 

3 

2438 

2494 

255i 

2608 

2665 

272I 

56 

3 

32o3 

3261 

3320 

3379 

3437 

3496 

56 

4 

2778 

2835 

2892 

2949 

3oo5 

3o62 

55 

4 

3555 

36i3 

3672 

373i 

3789*3848 

55 

5 

3n9 

3:76 

3233 

329o 

3346 

34o3 

54 

5 

39o7 

3965 

4o24 

4o83 

4:42 

4200 

54 

e 

346o 

35i7 

3574 

363i 

3687 

3744 

53 

6 

43i8 

4377 

4435 

4494 

4553 

53 

7 

38oi 

3858 

39i5 

3972 

4029 

4o86 

52 

7 

4612 

467o 

4?29 

4788 

4847 

4906 

52 

8 

4i43 

4199 

4256 

43i3 

437o 

4427 

5i 

8 

4964 

5o23 

5o82 

5i4i 

52OO 

5258 

5i 

9 

4484 

454i 

4598 

4655 

4712 

4769 

5o 

9 

53i7 

5376 

5435 

5494 

5553 

56i2 

5o 

10 

10.354826 

4883 

494o 

4997 

5o54 

5m 

49 

10 

io.37567o 

5729 

5788 

5847 

59o6 

5965 

49 

ii 

5i68 

5225 

5282 

5339 

5396 

5453 

48 

ii 

6024 

6o83 

6142 

6200 

6259 

63i8 

48 

12 

55io 

5567 

5624 

568i 

5738 

5795 

47 

12 

6377 

6436 

6495 

6554 

66i3 

6672 

47 

i3 

5852 

59o9 

5966 

6o23 

6080 

6137 

46 

i3 

673i 

679o 

6849 

69o8 

6967 

7026 

46 

i4 

6i94 

625i 

63o9 

6366 

6423 

648o 

45 

i4 

7o85 

7i44 

7203 

-726,2 

7321 

738o 

45 

i5 

6537 

6594 

665i 

6708 

6765 

6823 

44 

i5 

7439 

7498 

7557 

76i6 

7675 

7734 

44 

16 

6880 

6937 

6994 

7o5i 

7io8 

7166 

43 

16 

7793 

7853 

79I2 

797i 

8o3o 

8089 

43 

i7 

7223 

7280 

7337 

7394 

745i 

75o9 

42 

17 

8i48 

8207 

8266 

8325 

8384 

8444 

42 

18 

7566 

7623 

7680 

7737 

7795 

7852 

4i 

18 

85o3 

8562 

8621 

8680 

8739 

8799 

4i 

r9 

79°9 

7906 

8024 

8081 

8i38 

8i95 

4o 

i9 

8858 

8917 

8976 

9o35 

9o94 

9i54 

4o 

20 

io.358253 

83io 

8367 

8425 

8482 

8539 

39 

20 

io.3792i3 

9272 

933i 

939o 

945o 

95o9 

39 

21 

8596 

8654 

8711 

8768 

8826 

8883 

38 

21 

9568 

9627 

9687 

9746 

98o5 

9864 

38 

22 

894o 

8998 

9o55 

9II2 

91-70 

9227 

37 

22 

9924 

9983 

..42 

.  IO2 

.161 

.220 

37 

23 

9284 

9342 

9399 

9456 

957i 

36 

23 

10.380280 

o339 

0398 

o457 

o5i7 

o576 

36 

24 

9629 

9686 

9743 

98oi 

9858 

99l6 

35 

24 

o636 

o695 

0754 

0814 

o873 

0932 

35 

25 

9973 

.  .3i 

..88 

.i45 

.203 

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34 

25 

0992 

io5i 

IIIO 

I  I7O 

I229 

I289 

34 

26 

io.36o3i8 

o375|o433 

o49o 

o548 

o6o5 

33 

26 

1  348 

1407 

i467 

i526 

i586 

1  645 

33 

27 

o663 

0720 

0778 

o835 

0893 

o95o 

32 

27 

I7o5 

1764 

1824 

i883 

i943 

2OO2 

32 

28 

1008 

io65 

1123 

1180 

1238 

I295 

3i 

28 

2061 

2121 

2180 

2240 

2299 

2359 

3i 

29 

i353 

i4io 

i468 

i525 

i583 

i64i 

3o 

29 

2418 

2478 

2538 

2597 

2657 

2716 

3o 

3o 

io.36i698 

1756 

i8i3 

1871 

1928 

i986 

29 

3o 

io.382776 

2835 

2895 

2954 

3oi4 

3o74 

29 

3i 

2o44 

2IOI 

2l59 

2217 

2274 

2332 

28 

Si 

3i33 

3i93 

3252 

33i2 

3372 

343i 

28 

32 

2389 

2447 

25o5 

2562 

2620 

2678 

2-7 

32 

349i 

355o 

36io 

367o 

3729 

3789 

27 

33 

2735 

2793 

285i 

2905 

2966 

3024 

26 

33 

3849 

39o8 

3968 

4028 

4o87 

4i47 

26 

34 

3o8i 

3i97 

3255 

33i2 

337o 

25 

34 

4207 

4266 

4326 

4386 

4445 

45o5 

25 

35 

3428 

3486 

3543 

36oi 

3659 

37i7 

24 

35 

4565 

4625 

4684 

4744 

48o4 

4864 

24 

36 

3774 

3832 

389o 

3948 

4oo5 

4o63 

23 

36 

4923 

4983 

5o43 

5io3 

5i62 

5222 

23 

37 

4l2I 

4179 

4237 

4294 

4352 

44io 

22 

37 

5282 

5342 

5402 

546  1 

552i 

558i 

22 

38 

4468 

4526 

4584 

464i 

4699 

4757 

21 

38 

564i 

5701 

576i 

5820 

588o 

594o 

21 

39 

48i5 

4873 

493i 

4989 

5o46 

5io4 

20 

39 

6000 

6060 

6120 

6180 

6240 

0299 

2O 

4o 

io.365i62 

522O 

5278 

5336 

5394 

5452 

I9 

4o 

io.386359 

64i9 

6479 

6539 

6599 

6659 

19 

4i 

55IO 

5568 

5626 

5684 

574i 

5799 

18 

4i 

6719 

6779 

6839 

6899 

6959 

7019 

18 

42 

5857 

59I5 

5973 

6o3i 

6089 

6i47 

17 

42 

7079 

7i39 

7i99 

7259 

73i9 

7379 

'7 

43 

62o5 

6263 

632i 

6379 

6437 

6495 

16 

43 

7439 

7499 

7559 

76l9 

7679 

7739 

16 

44     6553 

6611 

6669 

6727 

6785 

6843 

:5 

44 

7799 

7859 

79i9 

7979 

8o39 

8o99 

i5 

45 

69oi 

696o 

7oi8 

7076 

7i34 

7I92 

i4 

45 

8i59 

82I9 

82?9 

8339 

8399 

846o 

i4 

46 

725o 

73o8 

7366 

7424 

7482 

754o 

i3 

46 

8520 

858o 

864o 

8700 

876o 

8820 

13 

47 

7598 

7657 

77i5 

7773 

783i 

7889 

12 

4? 

8880 

894i 

9ooi 

9o6i 

9I2I 

9181 

12 

48 

7947 

8oo5 

8o64 

8122 

8180 

8238 

I  I 

48 

9241 

93o2 

9362 

9422 

9482 

9542 

II 

49 

8296 

8354 

84i3 

84?i 

8529 

8587 

10 

49 

9603 

9663 

9723 

9783 

9844 

99o4 

10 

5o 

10.368645 

87o4 

8762 

8820 

8878 

8937 

9 

5o 

io.389964 

..24 

..85 

.i45 

.205 

.265 

9 

5i 

8995 

9o53 

9m 

9170 

9228 

9286 

8 

5i 

io.39o326 

o386 

o446 

o5o7 

o567 

0627 

8 

5a 

9344 

94o3 

946i 

95i9 

9578 

9636 

7 

52 

0688 

o748 

0808 

0869 

o929 

o989 

7 

53 

9694 

9753 

9811 

9869 

9927 

9986 

6 

53 

io5o 

IIIO 

I  I7O 

I23l 

1291:1352 

6 

54 

10.370044 

oio3 

0161 

0219 

0278 

o336 

5 

54 

l4l2 

1472 

i533 

i593 

i654 

i7i4 

5 

55 

o394 

o453 

o5n 

0569 

0628 

0686 

4 

55 

i775 

i835 

i895 

1956 

2016 

2077 

4 

56 

0745 

o8o3 

0862 

0920 

o978 

io37 

3 

56 

2l37 

2I98 

2258 

2319 

2379 

2440 

3 

57 

io95 

n54 

1212 

1271 

i329 

i388 

2 

07 

25oo 

256i 

2621 

2682 

2803 

3 

58 

1  446 

i5o4 

i563 

1621 

1680 

i738 

I 

58 

286312924 

2985 

3o45 

3io6 

3i66 

J 

59 

1797 

i855 

1914 

i972|2o3i 

2090 

O 

59 

3227{32873348 

3409 

3469 

353o 

O 

60" 

50"   40"  |  30"  i  20"   10"  | 

60"     50"   40"  1  30"  |  20"   10"  }  „. 

Co-tangent  of  23  Degrees. 

i 

Co-tangent  of  22  Degrees.    '  2 

C  i"  2"  3"  4"  5"  6"  7"  8"  9" 
PartJ  6  12  17  23  29  35  40  46  52 

P  Parti  l"  2//  3//  4"  '"  6"  7"  8'  n" 
Pj  6  12  18  24  30  3G  42  4S  51 

L  o  G  A  R  i  T  ii  M  r  c    SINES. 


1 

1 

Sine  of  68  Degrees. 

a       Sine  of  69  Degrees. 

ss 

0"     10" 

20"  !  30"   40" 

50"  1 

9     0"     10" 

1 

20"  |  30" 

40" 

50" 

o 

i 

o,.967i66|7i74|7i83  719117200 
7217  7225  7234  7242  7251 

7208 

?259 

59 

58 

o 
I 

9  970152 

O20O 

0160 
0208 

0168 
O2l6 

oi76 
0224 

0184 
0233 

OI92 

59 

58 

2J      7268 

7276 

7285  7293  7302 

73io 

57 

2 

0249 

0257 

0265 

0273 

0281 

O289 

57 

3 

73i9 

7327 

733673447353 

736i 

56 

3 

0297 

o3o5 

o3i3 

0321 

o329 

o337 

56 

4 

7370 

737873877395 

74o4 

74l2 

55 

4 

o345 

o353 

o36i 

o37o 

o378 

o386 

55 

5 

7421 

7437  7446 

7454 

7463 

54 

K 

o394 

O4O2 

o4io 

o4i8 

0426 

o434 

5J 

6 

7471 

7480 

7488  7497 

75o5 

75i4 

53 

6 

o442 

o45o 

o458 

o466 

o474 

0482 

53 

7 

7522 

753i 

7539754? 

7556 

7564 

52 

7 

o49o 

o498 

o5o6 

o5i4 

0522 

o53o 

5a 

8 

7573 

758i 

75907598 

7607 

76i5 

5i 

8 

o538 

o546 

o554 

o562 

o57o 

o578 

5i 

9 

•7624 

7632 

7640  7649 

7657 

7666 

5o 

9 

o586 

o594 

o6o3 

0611 

o6i9 

0627 

5o 

JO 

9.967674 

7683 

7691  7699 

7708 

7716 

49 

10 

9  .970635 

o643 

o65i 

o659 

o667 

o675 

49 

ii 

7725 

7733  7742  7750 

7758 

7767 

48 

ii 

o683 

o69i 

0699 

o7o7 

o7i5 

0723 

48 

12 

7775 

7784 

77937801 

78o9 

7817 

47 

12 

o73i 

o739 

0747 

o755 

o763 

0771 

47 

i3 

•7826 

7834 

7843  7851 

7859 

7868 

46 

i3 

0779 

o787 

o795 

o8o3 

0811 

08  1  9 

46 

i'4 

7876 

7885 

7893  7901 

79io 

79i8 

45 

i4 

0827 

o835 

0842 

o85o 

o858 

0866 

45 

i5 

7927 

7935 

7943  7952 

796o 

7069. 

44 

i5 

0874 

0882 

0890 

o898 

o9o6 

o9i4 

44 

16 

7977 

7985 

7994  8002 

80  1  1 

8019 

43 

16 

O922 

o93o 

o938 

o946 

o954 

00,62 

43 

17 

8027 

8o36  8o44  8o53 

8061 

8069 

42 

17 

°97° 

0978 

0986 

o994 

1  002 

1010 

42 

18 

8o78 

8o868o948io3 

8xxi 

8120 

4i 

18 

1018 

1026 

io34 

1042 

io5o 

io58 

4i 

I9 

8128 

8i368i458i53 

8161 

8170 

4o 

19 

1066 

1073 

1081 

io89 

io97 

no5 

4o 

20 

9.968i78 

818781968203 

8212 

8220 

39 

20 

9.97iii3 

II2I 

1129 

il37 

n45 

n53 

39 

21 

8228 

8237 

82458253 

8262 

8270 

38 

21 

1  161 

1169 

1177 

n85 

n93 

1  200 

38 

22 

8278 

82878295|83o3 

83i2 

8320 

37 

22 

1208 

1216 

1224 

1232 

1240 

1248 

37 

23 

8329 

8337  8345  8354 

8362 

8370 

36 

23 

1256 

1264 

1272 

1280 

1288 

I296 

36 

24 

8379 

8387 

8395  84o4 

8412 

8420 

35 

24 

i3o3 

i3n 

iSig 

l327 

i335 

1  343 

35 

25 

8437  8445  8454 

8462 

8470 

34 

25 

i35i 

i359 

i367 

i375 

i383 

i39o 

34 

26 

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8487849585o3 

85i2 

8520 

33 

26 

i398 

i4o6 

i4i4 

1422 

i43o 

143833 

27 

8528 

853785458553 

8562 

857o 

32 

27 

1  446 

i454 

1462 

i469 

i477 

i485 

32 

28 

8578 

8587 

8595 

86o3 

8612 

8620 

3i 

28 

i493 

i5oi 

1509 

i5i7 

i525 

i532 

3i 

29 

8628 

8636  8645 

8653 

8661 

8670 

3o 

29 

i54o 

1  548 

i556 

1  564 

l572 

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3o 

3o 

9.968678 

8686  8694 

87o3 

8711 

8719 

29 

3o 

9.971588 

i595 

i6o3 

1611 

1619 

1627 

29 

3  1 

8728 

87368744 

8752 

8761 

8769 

28 

3i 

i635 

i643 

i65i 

i658 

1666 

1674 

28 

32 

8777 

8786 

870.4 

8802 

8810 

8819 

27 

32 

1682 

i69o 

1698 

I7o6 

I7i3 

1721 

27 

33 

882? 

8835 

8844 

8852 

8860 

8868  26 

33 

1729 

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1745 

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i76i 

1768 

26 

34 

8877 

8885 

8893 

89oi 

89io 

891825 

34 

1776 

i784 

1792 

1800 

1808 

i8i5 

20 

35 

8926 

8934  8943 

895i 

8959 

8967 

24 

35 

1823 

i83i 

i839 

i847 

i855 

1862 

24 

36 

8976 

8984  8992 

9ooo 

9oo9 

9017 

23 

36 

1870 

i878 

1886 

i894 

1902 

I9o9 

23 

37 

9O25 

9o33  9o42 

9o5o 

9o58 

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22 

37 

1917 

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I933 

i94i 

1949 

i956 

22 

38 

9°75 

9o83  9o9i 

9°99 

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9116 

21 

38 

1964 

I972 

i98o 

i988 

i995 

2003 

21 

39 

9I24 

9i32  9i4i 

9i49 

9i57 

9i65 

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39 

2011 

2OI9 

2027 

2034 

2042 

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9.969i73 

9l82  9I90 

9i98 

9206 

9215 

19 

4o 

9.972O58 

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2073 

2081 

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2097 

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4i 

9223 

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9239 

9247 

92569264 

18 

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2IO5 

21  12 

2120 

2128 

2i36 

2144 

18 

42 

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928o9288 

9297 

93o5 

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17 

42 

2l5l 

2l59 

2167 

2I75 

2i83 

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17 

43 

932I 

9329  9338 

9346 

9354 

9362 

16 

43 

2198 

2206 

22l4 

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2229 

2237 

16 

44 

937° 

9379 

9387 

9395 

94o3 

9411 

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44 

2245 

2253 

2260 

2268 

2276 

2284 

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45 

942O 

9436 

9444 

9452 

9461 

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45 

2291 

2299 

2307 

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2322 

233o 

i4 

46 

946C) 

9477 

9485 

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gSio 

i3 

46 

2338 

2346 

2354 

236i 

2369 

2377 

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47 

95i8 

95269534 

955o 

9559 

12 

47 

2385 

2392 

2400 

2408 

24i6 

2423 

12 

48 

9567 

95759583 

959i 

9599 

9608 

II 

48 

2431 

2439 

2447 

2454 

2462 

247O 

II 

49 

96i6 

9624  9632 

9640 

9648  9657 

10 

49 

2478 

2485 

2493 

2501 

25oS 

25i6 

IO 

5o 

9,969665 

9673968i 

9689 

9697 

97°5 

9 

5o 

9.972524 

2532 

2539 

2547 

2555 

2563 

9 

5i 

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9722 

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9738 

9746  9754 

8 

5! 

2570 

2578 

2586 

2593 

2601 

26o9 

8 

52 

9762 

977i  9779 

9787 

9795  98o3 

7 

52 

2617 

2624 

2632 

2640 

2648 

2655 

7 

53 

9811 

9819 

9828 

9836 

984498*2 

6 

53 

2663 

2671 

2678 

2686 

260.4 

27OI 

6 

54 

986o 

98689876 

9884 

9893  9901 

5 

54 

2709 

2717 

2725 

2732 

2740 

2748 

5 

55 

99°9 

99179925 

9933 

994i'y949 

4 

55 

2755 

2763 

277I 

2778 

2786 

2794 

4 

56 

9957 

9966  9974 

9982 

999°9998 

3 

56 

2802 

2809 

281-7 

2825 

2832 

2840 

3 

57 

9,970006 

OOl4  0022 

oo3o 

oo38  0047 

2 

57 

2848 

2855  2863 

28-71 

2878  2886 

2 

58 

oo55 

oo63  0071 

0079 

0087  oo95 

I 

58 

2894 

2901  2909  2917 

2924!2032 

I 

59 

oio3 

OIII 

01  19  0127 

oi36  oi44 

0 

59 

2940  2947  2955  2963  297o!2978 

0 

fitv   j  50" 

40"   30"   20"   10" 

. 

60"     50"   40"   30"   20"   10"   =. 

Co-sine  of  21  Degrees. 

1 

Co-sine  of  20  Degrees. 

P  Parti  l"  2"  3" 

4"  5"  6"  7"  8"  9" 

D  „  A  V  2/;  3"  4''  5"  6"  7"  8"  9"  , 

*")  1   2   2 

345677       ll">  122345667 

LOGARITHMIC    TANGENTS. 


03 


£ 

Tangent  of  68  Degrees. 

d 

a 

Tangent  of  69  Degrees. 

s 

0"      10"  |  20"  |  30" 

40"   50"  7] 

2 

0"     |  10"  |  20" 

30" 

40" 

50" 

1 

o 

10.393590  365II37I2 

3772 

3833 

3894 

59 

0 

io.4i5823 

5886 

5948 

6011 

6074 

6i37 

591 

I 

3954 

4oi5 

4o76 

4i36 

4i97 

4258 

58 

i 

6200 

6263 

6326 

6389 

6452 

65i5 

58 

2 

43  1  8 

4379 

444o 

45oo 

456i 

4622 

57 

2 

6578 

664i 

67o4 

6767 

683o 

6893 

57 

<j 

4683 

4743 

48o4 

4865 

4926 

4986 

56 

3 

6956 

7O2O 

7083 

7i46 

7209 

7272 

56 

4 

5o47 

5io8 

5i6g 

5229 

5290 

535i 

55 

4 

7335 

7398 

746i 

7524 

7587 

765o 

55 

5 

54i2 

5473 

5533 

5594 

5655 

57i6 

54 

5 

77i4 

7777 

784o 

79°3 

7g66 

8020. 

54 

8 

5777 

5838 

5898 

5959 

6020 

6081 

53 

6 

8093 

8i56 

8219 

8282 

8345 

84o9 

53 

7 

6142 

6203 

6264 

6325 

6385 

6446 

52 

7 

8472 

8535 

8598 

8661 

8725 

8788 

52 

8 

65o7 

6568 

6629 

669o 

675i 

6812 

5i 

8 

885i 

89i4 

8978 

9041 

9104 

9168 

5i 

9 

6873 

6934 

6995 

7o56 

7117 

7178 

5o 

9 

9231 

9294 

9358 

9421 

9484 

9547 

5o 

10 

10.397239 

73oo 

736i 

7422 

7483 

7544 

49 

10 

10.419611 

9674 

9738 

9801 

9864 

9928 

49 

ii 

76o5 

7666 

7727 

7788 

7849 

79io 

48 

ii 

9991 

..54 

.118 

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.245 

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48 

12 

7971 

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8093 

8i54 

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8276 

47 

12 

10.4203-71 

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0625 

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47 

i3 

8337 

8399 

846o 

852i 

8582 

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46 

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46 

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45 

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n33 

1197 

1260 

1324 

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i45i 

45 

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9071 

9l32 

9194 

9255 

93i6 

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44 

i5 

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i64i 

I7o5 

1769 

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44 

16 

9438 

95oo!956i 

9622 

9683 

9744 

43 

16 

1896 

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2023 

2086 

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22l4 

43 

*7 

9806 

98679928 

9989 

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.  112 

42 

*7 

2277 

234l 

24o5 

2468 

2532 

2696 

42 

18 

10.400173 

0235 

0296 

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4i 

id 

2659 

2723 

2787 

285o 

29l4 

2978 

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3io5 

3i69 

3233 

3296 

336o 

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20 

10.400909 

0971 

1032 

io93 

n55 

1216 

3.9 

20 

10.423424 

3488 

355; 

36i5 

3679 

3743 

39 

21 

1278 

1339 

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1462 

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i585 

38 

21 

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3934 

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4062 

4126 

38 

22 

1  646 

1707 

1769 

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l892 

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3? 

22 

4190 

4253 

43i7 

438i 

4445 

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37 

23 

20l5 

2076 

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21  99 

2261 

2322 

36 

23 

4573 

4637 

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4764 

4828 

4892 

36 

24 

s384 

2445 

2507 

2568 

263o 

269I 

35 

24 

4956 

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5212 

5276 

35 

25 

2753 

28i5 

2876 

2938 

2999 

3o6i 

34 

25 

534o 

54o4 

5468 

5532 

5596 

566o 

34 

26 

3l22 

3i84 

3246 

33o7 

3369 

343o 

33 

26 

5724 

5788 

5852 

59i6 

598o 

6044 

33 

27 

3492 

3554 

36i5 

3677 

3739 

38oo 

32 

2H 

6108 

6l72 

6236 

63oo 

6364 

6429 

32 

28 

3862 

3924 

3985 

4o47 

4109 

4i7o 

3i 

28 

6493 

6557 

6621 

6685 

6749 

68i3 

3i 

29 

4232 

4294 

4356 

44i7 

4479 

454i 

3o 

29 

6877 

694i 

7oo6 

7o7o 

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3o 

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10.404602 

4664 

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4788 

485o 

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29 

3o 

10.427262 

7327 

739i 

7455 

75i9 

7583 

29 

3i 

4973 

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5097 

5i58 

5220 

5282 

28 

3i 

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7776 

784o 

79o5 

7969 

28 

32 

5344 

54o6 

5468 

5529 

559i 

5653 

27 

32 

8o33 

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8162 

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820,0 

8355 

27 

33 

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5839 

59oi 

5962 

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33 

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26 

34 

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6210 

6272 

6334 

6396 

25 

34 

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8869 

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0,062 

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25 

35 

6458 

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6582 

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24 

35 

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9320 

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24 

36 

6829 

6891 

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23 

36 

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23 

3? 

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745o 

75l2 

22 

37 

9965 

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22 

38 

7574 

7636 

7698 

7760 

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7884 

21 

38 

io.43o352 

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0610 

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21 

39 

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8008 

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20 

39 

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0868 

0933 

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1062 

2O 

4o 

io.4o83i9 

838i 

8443 

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19 

4o 

IO.43lI27 

n9i 

1256 

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4i 

8692 

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1  8 

4i 

,   i5i4 

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1  644 

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18 

42 

go65 

9127 

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42 

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2097 

2161 

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17 

43 

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95oi 

9563 

9625 

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16 

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229I 

2356 

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2485 

255o 

26i5 

16 

44 

9812 

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9937 

9999 

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44 

2680 

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2809 

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2939 

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45 

10.410186 

0248 

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45 

3o68 

3i33 

3198 

3263 

3328 

3393 

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46 

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1871 

1934 

i996 

10 

49 

4637 

4692 

4757 

4822 

4887 

4952 

10 

5o|io.  412069 

2121 

2184 

2246 

2309 

2371 

9 

5o 

io.435oi7 

5o82 

5i47 

5212 

5277 

5342 

9 

5-i 

2434 

2497 

2559 

2622 

2684 

2747 

8 

5i 

54o7 

5473 

5538 

56o3 

5668 

5733 

8 

62 

2810 

2872 

2935 

2997 

3o6o 

3i23 

7 

52 

5798 

5863 

5929 

5994 

6o59 

6124 

7 

53 

54 

3i85 
356i 

3248 
3624 

33ii 
3687 

3373 
3749 

3436 
33i2 

3499 

3875 

6 
5 

53 

54 

6i89 
658i 

6254 
6646 

6320 
67n 

6385 
6776 

645o 
6842 

65i5 
69o7 

6 

5 

55 

3938 

4ooo 

4o63 

4126 

4i89 

425i 

4 

55 

6972 

7o37 

7io3 

7168 

7233 

7299 

4 

56 

43i4 

4377 

444o 

45o2 

4565 

4628 

3 

56 

-  7364 

•7429 

7495 

7660 

7625 

769i 

3 

57 

4691 

4754 

48i7 

4879 

4942 

5oo5 

2 

57 

7756 

7822 

7887 

7952 

8018  8o83 

2 

58 

5o68 

5i3i 

5i94 

5256 

5319 

5382 

I 

58 

8i49 

8214 

8279 

8345 

84108476 

I 

59 

5445 

55o8 

557i 

5634 

5697 

6760 

0 

59 

854i 

86o7 

8672 

8738 

88o38869 

O 

60"      50"  |  40" 

30" 

20" 

10" 

d 

60" 

50" 

40" 

30-'' 

20"   10" 

~T 

Co  -tangent  of  21  Degrees. 

1 

Co-tangent  of  20  Degrees. 

1 

p  „   <  1"  2"  3"  4"  5"  6"  7"  8"  9" 

p  p  .(  1"  2"  3"  4/;  5"  6"  7"  8"  9" 

5%  6  12  19  25  31  37  43  49  56 

irtf  6  13  19  26  32  39  45  51  58 

LOGARITHMIC    SINES. 


d 

Sine  of  70  Degrees. 

a 

Sine  of  7  1  Degrees. 

& 

0" 

10" 

20" 

30" 

40" 

50" 

2 

0" 

10" 

20" 

30" 

40" 

50" 

o 

9.  9.729.86 

2993 

3ooi 

3oo9 

3oi6 

3021 

59 

0 

9.  0.75670 

5677 

5685 

5692 

5699 

57o6 

59 

I 

3o32 

3o47 

3o55 

3o62 

3O70 

58 

I 

5714 

5721 

5728 

5735 

5743 

575o 

58 

2 

3078 

3o85 

3o93 

3ioi 

3io8 

3n6 

57 

2 

5757 

5764 

577i 

5779 

5786 

5793 

57 

3    3i24 

3i3i 

3i39 

3i46 

3i54 

3i62 

56 

3 

58oo 

58o8 

58i5 

5822 

5829 

5837 

56 

4    3i69 

3i77 

3i85 

3i92 

3200 

3208 

55 

4 

5844 

585i 

5858 

5865 

5873 

588o 

55 

5 

32i5 

3223 

323o 

3238 

3246 

3253 

54 

5 

5887 

5894 

59oi 

59o9 

59i6 

5923 

54 

6 

326i 

3269 

3^76 

3284 

329I 

3299 

53 

0 

593o 

5938 

5945 

5952 

5959 

5966 

53 

7 

33o7 

33i4 

3322 

333o 

3337 

3345 

52 

7 

5974 

598i 

5988 

5995 

6002 

6010 

52 

8 

3352 

336o 

3368 

3375 

3383 

339o 

5i 

8 

6017 

6024 

6o3i 

6o38 

6o46 

6o53 

5i 

9 

3398 

34o6 

34i3 

342i 

3428 

3436 

5o 

9 

6060 

6067 

6074 

6081 

6o89 

6o96 

5o 

10 

9.973444 

345i 

3459 

3466 

3474 

3482 

49 

10 

9.976io3 

6110 

6117 

6i25 

6i32 

6i39 

49 

ii 

3489 

3497 

35o4 

35i2 

35i9 

35u7 

48 

ii 

6i46 

6i53 

6160 

6168 

6i75 

6182 

48 

12 

3535 

3542 

355o 

3557 

3565 

3572 

47 

12 

6i89 

6i96 

62o3 

6211 

6218 

6225 

47 

i3 

358o 

3588 

3595 

36o3 

36io 

36i8 

46 

i3 

6232 

6239 

6246 

6254 

6261 

6268 

46 

i4 

3625 

3633 

364i 

3648 

3656 

3663 

45 

i4 

6275 

6282 

6289 

620.6 

63o4 

63n 

45 

i5 

367i 

3678 

3686 

3694 

3701 

37o9 

44 

i5 

63i8 

6325 

6332 

6339 

6347 

6354 

44 

.  16 

3716 

3724 

373i 

3739 

3746 

3754 

43 

16 

636i 

6368 

6375 

6382 

6389 

6396 

43 

17 

376i 

3769 

3777 

3784 

3792 

3799 

42 

I7 

64o4 

64n 

64i8 

6425 

6432 

6439 

42 

18 

3807 

38i4 

3822 

3829 

3837 

3844 

4i 

18 

6446 

6454 

646  1 

6468 

6475 

6482 

4  1 

'9 

3852 

3859 

3867 

3875 

3882 

389o 

4o 

i9 

6489 

6496 

65o3 

65io 

65i8 

6525 

4o 

20 

9.973897 

39o5 

39I2 

3920 

3927 

3935 

39 

20 

9.976532 

6539 

6546 

6553 

656o 

6567 

39 

21 

395o 

39573965 

3972 

398o 

38 

21 

6574 

6582 

6589 

6596 

66o3 

6610 

38 

22 

3987 

3995 

4002  4oio 

4017 

4o25 

37 

22 

6617 

6624 

663i 

6638 

6646 

6653 

37 

23 

4o32 

4o4o 

4o47i4o55 

4062 

4070 

36 

23 

6660 

6667 

6674 

6681 

6688 

6695 

36 

24 

4o77 

4o85 

4092  4lOO 

4107 

4n5 

35 

24 

6702 

67o9 

6716 

6723 

673i 

6738 

35 

25 

4l22 

4i3o 

4i37!4i45 

4i52 

4i6o 

34 

25 

6745 

6752 

6759 

6766 

6773 

678o 

34 

26 

4l67 

4i75 

4182  4i9o 

4197 

42o5 

33 

26 

6787 

670.4 

6801 

6808 

68i5 

6823 

33 

27 

4212 

4220 

42274235 

4242 

425o 

32 

27 

683o 

6837 

6844 

685i 

6858 

6865 

32 

28 

4257 

4264 

42724279 

4287 

4294 

3i 

28 

6872 

6879 

6886 

6893 

69oo 

69o7 

3i 

29 

43o2 

43o9 

43i7  4324  4332 

4339 

3o 

29 

69i4 

692I 

60.28 

6935 

6942 

695o 

3o 

309.974347 

4354 

436i!4369!4376 

29 

3o 

9.976957 

6964 

697i 

6978 

6985 

6992 

29 

3:|    439i 

4399 

44o644i4 

4421 

4428 

28 

3i 

6999 

7006 

7013 

7O2O 

7027 

7o34 

28 

32 

4436 

4443 

445i  4458 

4466 

44?3 

27 

32 

704  1 

7o48 

7o55 

7o62 

7o69 

7o76 

27 

33 

448  1 

4488 

4495!45o3 

45io 

45i8 

26 

33 

7083 

7o9o 

7°97 

7104 

7in 

7118 

26 

34 

4525 

4533 

454o4547 

4555 

4562 

25 

34 

7i25 

7182 

7i39 

7i46 

7i53 

7i6o 

25 

35 

457o 

4577 

4585450.2 

4699 

4607 

24 

35 

7167 

7174 

7181 

7188 

7i95 

7202 

24 

36 

46i4 

4622 

46294636 

4644 

465i 

23 

36 

72O9 

7216 

7223 

7280 

7237 

7244 

23 

37 

4659 

4666 

46744681 

4688 

4696 

22 

37 

725l 

7258 

7265 

7272 

7279 

7286 

22 

38 

47o3 

47n 

47i8,4725 

4733 

474o 

21 

38 

7293 

7800 

73o7 

73i4 

732I 

7328 

21 

39 

4748 

4755 

47624770 

4777 

4784 

20 

39 

7335 

7^42 

7']49 

7356 

7363 

737o 

2O 

4o 

9-974792 

4799 

4807 

48i4 

4821 

4820. 

'9 

4o 

9-977377 

7  384 

739i 

7398 

74o5 

74l2 

I9 

4i 

4836 

4844 

485i 

4858 

4866 

4873 

18 

4i 

7426 

7433 

744o 

7447 

7454 

18 

4a 

488o 

4888 

4895 

49o3 

4910 

4917 

17 

42 

7461 

7468 

7475 

7482 

7489 

•7496 

17 

43 

4925 

4932 

4939 

4947 

4954 

4961 

16 

43 

75o3 

75io 

75i7 

7524 

753o 

7537 

16 

44 

4969 

4976 

4984 

4991 

4998 

5oo6 

i5 

44 

7544 

755i 

7558 

7565 

7572 

7579 

i5 

45 

5oi3 

5O2O 

5028 

5o35 

5o42 

5o5o 

i4 

45 

7586 

7593 

76oo 

7607 

76i4 

762I 

i4 

46 

5o57 

5o64 

5072 

5o79 

5o86 

5o94 

i3 

46 

•7628 

7635 

7642 

7648 

7655 

7662 

i3 

47 

5ioi 

5io8 

5n6 

5i23 

5i3o 

5i38 

12 

47 

7660. 

7676 

7683 

760.0 

7697 

77o4 

12 

48 

5i45 

5i52 

5i6o 

5i67 

5i74 

5i82 

II 

48 

7711 

7718 

7725 

7732 

7739 

7745 

II 

49 

5189 

5i96 

52o4 

5211 

52i8 

5226 

IO 

49 

775a 

7759 

7766 

7773 

778o 

7787 

10 

5o 

9.975233 

524o 

5248 

5255 

5262 

5270 

9 

5o 

9-977794 

7801 

78o8 

78i5 

782I 

7828 

9 

5i 

5277 

5284 

5292 

5299 

53o6 

53i3 

8 

5i 

7835 

7842 

7849 

7856 

7863 

787o 

8 

52 

532i 

5328 

5335 

5343 

535o 

5357 

7 

52 

7877 

7884 

789o 

7897 

79o4 

7911 

7 

53 

5365 

5372 

5379 

5386 

5394 

54oi 

6 

53 

79i8 

7925 

7932 

7939 

7946 

7952 

6 

54 

54o8 

54i6 

5423 

543o 

5437 

5445 

5 

54 

7959 

7966 

7973 

798o 

7987 

7994 

5 

55 

5452 

5459 

5467 

5474 

548  1 

5488 

4 

55 

8001 

8007 

8014 

8021 

8028 

8o35 

4 

'  56 

5496 

55o3 

55io 

55i8 

5525 

5532 

3 

56 

8042 

8049 

8o56 

8062 

8o69 

8o76 

3 

57 

5539 

5547 

5554 

556i 

5568 

5576 

2 

57 

8o83 

8o9o 

8o97 

8io4 

8110 

8117 

2 

53 

5583 

559o 

5598 

56o5 

56i2 

56i9 

I 

58 

8124 

8i3i 

8i38 

8i45 

8i52 

8i58 

I 

59 

5627 

5634 

564i 

5648 

5656 

5663 

0 

59 

8  1  65 

8172 

8i79 

8186 

8i93 

8i99 

O 

60"    |  50" 

40" 

30"  |  20" 

10" 

d 

60" 

50"  |  40" 

30"  |  20" 

10" 

ri 

Co-sine  of  19  Degrees. 

Co-sine  of  18  Degrees. 

i 

'  p  p   (  1"  2"  3"  4"  5"  6"  7"  8"  9" 

.(  l';  2"  3"  4''  5"  6"  7"  8"  9" 

P.lartJ  !]   2  3   4   4   5   6   7 

in  $1123   4   456   6 

LOGARITHMIC    TANGENTS. 


05 


1 

Tangent  of  70  Degrees. 

a 

Tangent  of  71  Degrees. 

2 

0" 

10" 

20" 

30" 

40" 

50" 

S 

0"     |  10" 

20"   30" 

40" 

5<y  | 

0 

10.438934 

9000 

9O65 

9i3i 

9196 

9262 

r 

O 

io.463o28 

3o97 

3i65 

3233 

33o2 

337o 

59 

I 

9327 

9393 

9458 

9524 

959° 

9655 

58 

I 

3439 

35o7 

3576 

3644 

37i3 

378i 

58 

2 

9721 

9786 

9852 

99i8 

9983 

..49 

5^7 

2 

385o 

39i8 

3987 

4o55 

4124 

4l92 

57 

3 

io.44oi  i5 

0180 

0246 

O3l2 

o377 

o443 

56 

3 

4261 

4329 

4398 

4466 

4535 

46  04 

56 

4 

oSog 

o574 

o64o 

0706 

o77i 

o837 

55 

4 

4672 

474i 

48o9 

4878 

4947 

5oi5 

55 

5 

0903 

0969 

io34 

IIOO 

1166 

1232 

54 

5 

5o84 

5i53 

5221 

529o 

5359 

5427 

54 

6 

1297 

i363 

1429 

i495 

i56i 

1627 

53 

6 

5496 

5565 

5633 

57O2 

577i 

584o 

53 

7 

1692 

i758 

1824 

i89o 

1956 

2022 

52 

7 

59o8 

5977 

6o46 

6n5 

6184 

6252 

52 

8 

2087 

2  1  53 

2219 

2285 

235i 

24l7 

5i 

8 

632i 

639o 

6459 

6528 

6596 

6665 

5i 

9 

£483 

2549 

26i5 

2681 

2747 

28l3 

5o 

9 

6734 

68o3 

6872 

694i 

7010 

7079 

5o 

10 

10.442879 

2945 

3on 

3o77 

3i43 

3209 

49 

10 

10.467147 

7216 

7285 

7354 

7423 

7492 

49 

ir 

3275 

334i 

34o7 

34?3 

3539 

36o5 

48 

ii 

756i 

763o 

7699 

7768 

7837 

79o6 

48 

12 

367i 

3737 

38o3 

3869 

3935 

4ooi 

47 

12 

7975 

8o44 

3n3 

8182 

825i 

8320 

47 

i3 

4067 

4i33 

4200 

4266 

4332 

4398 

46 

i3 

8389 

8458 

8527 

8597 

8666 

8735 

46 

M 

4464 

453o 

4596 

4663 

4729 

4795 

45 

i4 

88o4 

8873 

8942 

9OII 

9o8o 

9i5o 

45 

i5 

486i 

4927 

4994 

5o6o 

5i26 

5192 

44 

i5 

9219 

9288 

9357 

9426 

9496 

9565 

44 

16 

5259 

5325 

5391 

5457 

5524 

559o 

43 

16 

9634 

9703 

9772 

9842 

99n 

998o 

43 

*7 

5656 

5722 

5789 

5855 

5921 

5988 

4s 

!7 

10.470049 

on9 

0188 

0257 

0327 

o396 

42 

18 

6o54 

6120 

6187 

6253 

6320 

6386 

4i 

18 

o465 

o535 

0604 

o673 

o743 

0812 

4i 

'9 

6452 

65i9 

6585 

6652 

6718 

6784 

4o 

:9 

0881 

o95i 

IO2O 

io9o 

n59 

1228 

4o 

20 

io.44685i 

6917 

6984 

7o5o 

7117 

7i83 

39 

20 

10.471298 

i367 

i437 

i5o6 

i576 

1  645 

39 

21 

725o 

73i6 

7383 

7449 

75i6 

7582 

38 

21 

1715 

1784 

i854 

I923 

i993 

2062 

38 

22 

7649 

77i5 

7782 

7848 

7915 

7981 

37 

22 

2l32 

22OI 

2271 

234o 

2410 

2480 

37 

23 

8o48 

8n5 

8181 

8248 

83i4 

838i 

36 

23 

2549 

26l9 

2688 

2758 

2828 

2897 

36 

24 

8448 

85i4 

858i 

8647 

8714 

8781 

35 

24 

2967 

3o37 

3io6 

3i76 

3246 

33i5 

35 

25 

8847 

8914 

8981 

9o48 

9114 

9181 

34 

25 

3385 

3455 

3524 

3594 

3664 

3734 

34 

26 

9248 

93i4 

9381 

9448 

95i5 

958i 

33 

26 

38o3 

3873 

3943 

4oi3 

4082 

4i52 

33 

27 

9648 

9715 

9782 

9848 

9915 

9982 

32 

27 

4222 

4292 

4362 

4432 

45oi 

457i 

32 

28 

io.45oo49 

0116 

oi83 

0249 

o3i6 

o383 

3i 

28 

464i 

4711 

4781 

485i 

492I 

4991 

3i 

29 

o45o 

o5i7 

o584 

o65i 

0717 

0784 

3o 

29 

5o6o 

5i3o 

5200 

527O 

534o 

54io 

3o 

3o 

io.45o85i 

0918 

0985 

IO52 

1119 

1186 

29 

3o 

io.47548o 

555o 

5620 

569o 

576o 

583o 

29 

3i 

1253 

1320 

i387 

1454 

l52I 

1  588 

28 

3i 

5900 

597o 

6o4o 

6110 

6180 

625o 

28 

32 

j.655 

1722 

1789 

i856 

1923 

1990 

27 

32 

6320 

639i 

646  1 

653i 

6601 

667i 

2? 

33 

2057 

2124 

2191 

2258 

2325 

2392 

26 

33 

674i 

68  1  1 

8881 

6952 

7O22 

7o92 

26 

34 

2460 

2527 

2594 

2661 

2728 

2795 

25 

34 

7162 

7232 

7302 

7373 

7443 

75i3 

25 

35 

2862 

2929 

2997 

3o64 

3i3i 

3198 

24 

35 

7583 

7654 

7724 

7794 

7864 

7935 

24 

36 

3265 

3333 

34oo 

3467 

3534 

3602 

23 

36 

8oo5 

8o75 

8i46 

8216 

8286 

8357 

23 

3? 

3669 

3736 

38o3 

387i 

3938 

4oo5 

22 

37 

84a7 

8497 

8568 

8638 

87o8 

8779 

22 

38 

4072 

4i4o 

4207 

4274 

4342 

4409 

21 

38 

8849 

8920 

899o 

9o6i 

9i3i 

92OI 

21 

39 

4476 

4544 

46u 

4678 

4746 

48i3 

2O 

39 

9272 

9342 

94i3 

9483 

9554 

9624 

2O 

4o 

io.45488i 

4948 

5oi5 

5o83 

5i5o 

52i8 

J9 

4o 

10.479695 

9765 

9836 

99o6 

9977 

..47 

*9 

4i 

5285 

5353 

5420 

5488 

5555 

5623 

18 

4-i 

10.480118 

0189 

0259 

o33o 

o4oo 

o47i 

18 

42 

5690 

5758 

5825 

5893 

5960 

6028 

!7 

42 

o542 

0612 

o683 

o754 

0824 

o895 

I7 

43 

6095 

6i63 

623o 

6298 

6365 

6433 

16 

43 

0966 

io36 

1107 

1178 

1248 

i3i9 

16 

44 

65oi 

6568 

6636 

67o3 

6771 

6839 

i5 

44 

1390 

i46i 

i53i 

1602 

i673 

i744 

i5 

45 

6906 

6974 

7042 

7io9 

7177 

7245 

i4 

45 

1814 

i885 

i956 

2027 

2C98 

2l69 

i4 

46 

73l2 

738o 

7448 

75i5 

7583 

765i 

i3 

46 

2239 

23lO 

238i 

2452 

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7922 

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47 

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2736 

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2877 

2948 

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48 

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8i93 

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8329 

8397 

8464 

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48 

3090 

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3232 

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49 

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8668 

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10 

49 

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53 

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6 

53 

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5437 

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6 

54 

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5 

54 

565i 

5722 

5794 

5865 

5936 

6008 

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55 

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1116 

1184 

1253 

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55 

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6222 

6293 

6364 

6436 

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1389 

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1662 

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3 

56 

65o7 

6579 

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6722 

6793 

6864 

3 

57 

1798 

1867 

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2003 

2071 

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2 

57 

6936 

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7°79 

7i5o 

7222 

7293 

2 

58 

2208 

2276 

2345 

24i3 

2481 

255o 

I 

58 

7365 

7436 

75o8 

758o 

765i 

7T23 

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59 

2618 

2686 

2755 

2823 

2891 

296o 

O 

59 

7794 

7866 

7937 

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8o8l 

8:52 

(' 

60" 

50"  |  40"   30" 

20" 

10" 

a 

60"     50"   40"   30" 

20" 

10" 

a 

Co-tangerit  of  19  Degrees. 

S 

Co-tangent  of  18  Degrees. 

i 

p  p   <  1"  2"  3"  4"  5"  6"  7"  8"  9" 
irl\  7  13  20  27  33  40  47  54  60 

P  P-,rt5  l"   ~"  3//   4"   5"   6//  7//   8//  •*" 

I  7   14  21  28  35  42  49  56  §3 

LOGARITHMIC    SINES. 


.3 

Sine  of  72  Degrees. 

d 

Sine  of  73  Decrees. 

s 

0" 

10" 

20"  |  30"   40" 

50" 

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0" 

10" 

20'' 

30" 

40" 

50" 

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9.9782o6 

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58 

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57 

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3 

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4 

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55 

5 

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8438 

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54 

5 

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6 

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53 

6 

0827 

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53 

7 

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32 

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33 

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3 

2 

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2696  2702  27o9'27i5 
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58    2769|2775  2781  2787  2793:2799 
59    2805:28112817282428302836 

I 
O 

60" 

50" 

40"   30"  |  20"   10" 

. 

60"    |  50"   40"   30"   20"  :  10"^  f- 

Co-sine  of  17  Degrees. 

1 

Co-sine  of  16  Degrees. 

P  Port  S  l"   2"   3"   4"   5"   6"  ?"   8"  9" 

£112334556 

P  Part*1"  2"  3"  4"  5//  6"  7"  *"  9/ 
irt{  1   1   2   2   3   4   4  5   f 

LOGARITHMIC    TANGENTS. 


1  — 
d 

Tangent  of  72  Degrees. 

.5 

Tangent  of  73  Degrees. 

& 

0" 

10" 

20"  |  30" 

40" 

50" 

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0" 

10" 

20" 

30" 

40'   50" 

o 

10.488224 

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8654 

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58 

2 

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57 

2 

5565 

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5716 

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5867 

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3     95i5 

9587 

9659 

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9802 

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56 

3 

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6320 

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4 

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.  162 

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10.490378 

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54 

5 

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54 

6 

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1025 

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1169 

53 

6 

7379 

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7606 

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53 

7 

1241 

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1602 

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7833 

79°9 

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8061 

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53 

8 

1674 

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5i 

8 

8288 

8364 

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8592 

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9 

2107 

2179 

225l 

2323 

2395 

2468 

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9 

8743 

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10 

10.492540 

2612 

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2757 

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9275 

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9427 

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60" 

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30" 

20" 

10" 

d 

60" 

50" 

40" 

30" 

20" 

10" 

£ 

Co-tangent  of  17  Degrees. 

i 

Co-tangent  of  16  Degrees. 

X 

V  l»Tt$  1/;   2//   3//   4"   5//   6"   7//   8//   9// 

t  r/  2"  3"  4"  5"  6"  7"  8"  9" 

'   Jt{  7  15  22  20  37  44  51  59  66 

l.rart^  g  15  23  31  3J)  4G  51  62  7fl 

L.OGARITHMIC    SINES. 


jj 

Sine  of  74  Degrees. 

| 

Sine  of  75  Degrees. 

0" 

10" 

20"   30"   40" 

50" 

0" 

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40" 

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d 

60" 

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a 

Co-sine  of  15  Degrees. 

Co-sine  of  14  Degrees. 

f  1  //  nil   q// 

P.Part{\  \  I 

233455 

P  Part*1"  2"  3"  4"  #'  *""?"  8"  9" 
irt}  1   12233445 

LOGARITHMIC    TANGENTS. 


99 


1 

Tangent  of  74  Degrees. 

.a 

Tangent  of  75  Degrees. 

i 

0" 

10"  |  20 

30" 

40" 

50" 

s 

0" 

10" 

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5o45 

5i33 

i5 

45 

4424 

4507 

459o 

4673 

4756 

4839 

i4 

45 

5222 

53io 

5398 

5486 

5575 

5663 

i4 

46 

4922 

5oo5 

5o88 

5l72 

5255 

5338 

i3 

46 

575i 

584o 

5928 

6017 

6io5 

6i93 

i3 

4? 

542i 

55o4 

5587 

5670 

5754 

5837 

12 

4? 

6282 

637o 

6459 

6547 

6636 

6724 

12 

48 

5920 

6oo3 

6086 

6170 

6253 

6336 

I  I 

48 

68i3 

6901 

6990 

7078 

7167 

7256 

II 

49 

6420 

65o3 

6586 

6669 

6753 

6836 

10 

49 

7344 

7433 

7522 

7610 

7699 

7788 

IO 

5o 

10.566920 

7003 

7086 

7170 

7253 

7337 

9 

5o 

io.597876 

7965 

8o54 

8i43 

823i 

832C 

9 

Si 

7420 

75o4 

7587 

7671 

7754 

7838 

8 

5i 

84o9 

8498 

8587 

8675 

8764 

8853 

8 

5s 

7921 

8oo5 

8088 

8172 

8255 

8339 

7 

52 

8942 

9o3i 

9120 

9209 

9298 

9387 

7 

53 

8423 

85o6 

859o 

8674 

8757 

884i 

6 

53 

9476 

9565 

9654 

9743 

9832 

992I 

6 

54 

8925 

9008 

9092 

9176 

926o 

9343 

5 

54 

10.600010 

OIOO 

0189 

0278 

o367 

o456 

5 

55 

9427 

gSn 

9595j9679 

9763 

9846 

4 

55 

o545 

o635 

0724 

o8i3 

0902 

O992 

4 

56 

9930 

..i4 

..98  .182 

.266 

.35o 

3 

56 

1081 

I  I7O 

1260 

1349 

i438 

i528 

3 

57 

10.570434 

o5i8 

0602  0686 

0770 

o854 

2 

57 

161-7 

I7o6 

1796 

i885 

i975  2064 

2 

58 

o938 

1022 

1106 

1  190 

I274 

i358 

I 

58 

2i54 

2243 

2333 

2422 

25l2  2601 

r 

69 

1  442 

1527 

1611 

i695 

I779 

i863 

0 

59 

269I 

278l 

2870 

2960 

3o5o3i39 

o 

60" 

50"   40"   30" 

20"  1  10" 

c 

60"      50" 

40" 

30"  , 

20"   10" 

d 

Co-tangent  of  15  Degrees. 

•9 

g 

Co-tangent  of  14  Degrees. 

i 

P  P«  t5  r"  2"  3"  4"  5"  6"  7"  8"  9" 

p  Po  .5  1"  2"  3"  4"  5"  6"  7"  8"  9" 

VJ  8  16  25  33  41  49  57  65  74 

irt>  9  17  26  35  43  52  61  69  78 

100 


LOGARITHMIC    IS  i  N  E  s. 


pi 

B 

Sine  of  76  Decrees. 

.= 

Sine  of  77  Degrees. 

8 

0" 

10" 

20" 

30" 

40"   50" 

* 

0" 

10" 

20" 

30" 

40" 

50" 

o 

9.986904 

6909 

69i5 

6920 

6925 

693o 

59 

O 

9.988724 

8729 

8734 

8739 

8743 

8748 

59 

i 

6936 

6941 

6946 

695i 

6957 

6962 

58 

I 

8753 

8758 

8763 

8768 

8772 

8777 

58 

2 

6967 

6972 

6978 

6983 

6988 

6993 

57 

2 

8782 

8787 

8792 

8797 

8802 

8806 

57 

3 

6998 

7004 

7oo9 

7014 

-7019 

7025 

56 

3 

8811 

8816 

8821 

8826 

883i 

8835 

56 

4 

7o3o 

7035 

7o4o 

7045 

7o5i 

7o56 

55 

4 

884o 

8845 

885o 

8855 

8880 

8864 

55 

5 

7061 

7066 

7072 

7077 

7o82 

7o87 

54 

5 

8869 

8874 

8879 

8884 

8889 

8893 

54 

6 

7092 

7098 

7io3 

7108 

7n3 

7118 

53 

6 

8898 

89o3 

89o8 

89i3 

89i8 

8922 

53 

7 

7124 

7129 

71  34 

7i39 

7i44 

7i5o 

52 

7 

8927 

8932 

8937 

8942 

8946 

895i 

52 

8 

7i55 

7160 

7i65 

7i7o 

7i76 

7i8i 

5i 

8 

8956 

896i 

8966 

897o 

8975 

898o 

5i 

9 

7186 

7191 

7196 

72O2 

7207 

72I2 

5o 

9 

8985 

899o 

8994 

8999 

9oo4 

9009 

5o 

10 

9.987217 

7222 

7228 

7233 

7238 

7243 

49 

IO 

9.989oi4 

9oi8 

9O23 

9028 

9o33 

9o?S 

49 

ii 

7248 

7253 

7259 

7264 

-7269 

7274 

48 

ii 

9o42 

9o47 

9o52 

9o57 

9o62 

9o66 

48 

12 

7279 

7284 

7290 

7295 

73oo 

73o5 

47 

12 

9o7i 

9o76 

9o8i 

9o85 

9o9o 

9o95 

47 

i3 

73io 

73i5 

732I 

7326 

733i 

7336 

46 

i3 

9100 

9io5 

9I09 

9n4 

9n9 

9I24 

46 

i4 

734i 

7346 

7352 

7357 

7362 

7367 

45 

14 

9128 

9i33 

9i38 

9i43 

9148 

9l52 

45 

i5 

7372 

7377 

7383 

7388 

7393 

7398 

44 

i5 

9i57 

9l62 

9i67 

9i7i 

9176 

9181 

44 

16 

18 

74o3 
7434 
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74o8 

7439 
7470 

74i3  7419 
7444  7449 
7475  748o 

7454 
7485 

7429 
746o 
749o 

43 

42 

4i 

16 

17 

18 

9186 
9214 
9243 

9i9o9i95 
921919224 

92479252 

92OO 
9228 
9257 

9205 
9233 
9262 

9209 

9238 
9266 

43 

42 

4i 

19 

7496 

75oi 

75o6  75n 

75i6 

752I 

4o 

J9 

927I 

9276 

928l 

9285 

9290 

9295 

4o 

20 

9.987526 

753i 

7537  7542 

7547 

7552 

39 

20 

9.9893oo 

93o4 

93o9 

93i4 

93i8 

9323 

39 

21 

7557 

7562 

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7^77 

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38 

21 

9328 

9333 

9337 

9342 

9347 

9352 

38 

22 

7588 

7593 

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76i3 

37 

22 

9356 

936i 

9366 

937o 

9375 

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37 

23 

7618 

7623 

7628  7634 

7639 

7644 

36 

23 

9385 

9389 

9394 

9399 

94o3 

94o8 

36 

24 

7649 

7654 

7659  7664 

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7674 

35 

24 

94i  3 

94i7 

9422 

9427 

9432 

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35 

25 

7679 

7684 

7690  7695 

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7706 

34 

25 

944^ 

9446 

945o 

9455 

946o 

9464 

34 

26 

7710 

77i5 

77207725 

773o 

7735 

33 

26 

9469 

9474 

9479 

9483 

9488 

9493 

33 

27 

774o 

7745 

775o7756 

7766 

32 

27 

9497 

95o2 

95°7 

95n 

95i6 

952I 

32 

28 

7771 

7776 

778i  7786 

7791 

7796 

3  1 

28 

9525 

953o 

9535 

9539 

9544 

9549 

3i 

29 

7801 

7806 

7811  7816 

7821 

7826 

3o 

29 

9553 

9558 

9563 

9568 

9572 

9577 

3o 

3o 

9.987832 

7837 

7842  7847 

7852 

7867 

29 

3o 

9.989582 

9586 

959i 

9596 

96oo 

96o5 

29 

3i 

7862 

7867 

7872  7877 

7882 

7887 

28 

3i 

9610 

96i4 

96l9 

9628 

9628 

9633 

28 

32 

7892 

7897 

7902  7907 

7912 

79i7 

27 

32 

9637 

9642 

9647 

965i 

9666 

9661 

27 

33 

7922 

7927 

7932  7937 

7942 

7947 

26 

33 

9665 

967o 

9675 

9679 

9684 

9689 

26 

34 

7953 

7958 

7963  7968 

7973 

7978 

25 

34 

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9707 

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9710 

25 

35 

7983 

7988 

79937998 

8oo3 

8008 

24 

35 

9726 

973o 

9735 

974o 

9744 

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36 

8oi3 

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8o33 

8o38 

23 

36 

9749 

9753 

9758 

9763 

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9772 

23 

37 

8o43 

8o48 

8o538o58 

8o63 

8068 

22 

37 

9777 

978i 

9786 

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9795 

9800 

23 

38 

8o73 

8078 

8o83  8088 

8093 

8o98 

21 

38 

9804 

98o9 

98i4 

98i8 

9823 

9827 

21 

39 

8io3 

8108 

8n38n8 

8i23 

8128 

2O 

39 

9832 

9837 

984i 

9840 

985o 

9855 

2O 

4o 

9.988133 

8i38 

8i438i48 

8i53 

8i58 

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4o 

9.98986o 

9864 

9869 

9873 

9878 

9883 

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4  1 

8i63 

8168 

81738178 

8i83 

8188 

18 

4i 

9887 

9892 

9896 

99oi 

99o6 

9910 

18 

42 

8193 

8198 

8.2638208 

82i3 

8218 

17 

42 

99i5 

99i9 

9924 

9929 

9933 

9938 

17 

43 

8223 

8227 

8232 

8237 

8242 

8247 

16 

43 

9942 

9947 

9952 

9956 

996i 

9965 

16 

44 

8252 

8257 

8262 

8267 

8272 

8277 

i5 

44 

997° 

9974 

9979 

9984 

9988 

9993 

i5 

45 

8282 

8287 

8292 

82978802 

8307 

i4 

45 

9997 

.  .  .2 

.  .  .6 

..ii 

..16 

.  .20 

i4 

46 

83i2 

83i7 

8322 

83278332 

8337 

i3 

46 

9.990025 

OO29 

oo34 

oo38 

oo43 

oo48 

i3 

47 

8342 

8346 

835i 

8356836i 

8366 

12 

47 

oo52 

oo57 

006  1 

0066 

0070 

0075 

12 

48 

837i 

8376 

838i 

83868391 

8396 

II 

48 

°°79 

0084 

0088 

oo93 

0098 

OIO2 

II 

49 

84oi 

84o6 

84n 

84168420 

8425 

IO 

49 

OIO7 

OIII 

0116 

OI2O 

0125 

OI29 

10 

5o 

9.988430 

8435 

844o 

8445 

845o 

8455 

9 

5o 

9.99oi34 

oi38 

oi43 

oi48 

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0157 

9 

5i 

846o 

8465 

8470 

8475 

848o 

8484 

8 

5i 

0161 

0166 

OI7O 

oi75 

oi79 

oi84 

8 

52 

8489 

8494 

8499 

85o4 

8509 

85i4 

7 

52 

0188 

oi93 

oi97 

0202 

0206 

O2II 

7 

53 

85i9 

8524 

8529 

8534 

8538 

8543 

6 

53 

O2l5 

O22O 

0225 

O229 

0234 

0238 

6 

54 

8548 

8553 

8558 

8563 

8568 

8573 

5 

54 

0243 

0247 

O252 

O256 

0261 

0265 

5 

55 

8578 

8583 

8587 

8592 

8597 

8602 

4 

55 

O27O 

0274 

0279 

0283 

0288 

O292 

4 

56 

8607 

8612 

8617 

8622 

8626 

863i 

3 

56 

O297 

o3oi 

o3o6 

o3io 

o3i5 

o3i9 

3 

57 

8636 

864i 

8646 

865i 

8656 

8661 

2 

57 

0324 

o328 

o333 

0337 

0342 

o346 

2 

•58 

8666 

8670 

867518680 

8685 

869o 

I 

58 

o35i 

o355 

o36o 

o364 

o369 

0373 

I 

'  69 

8695 

8700 

870487098714 

87i9 

0 

59 

o378 

o382 

o386 

o39i 

o395 

o4oo 

O 

60" 

50" 

40"  1  30"  |  20" 

10" 

d 

60"    |  50" 

40" 

30"   20"   10" 

d 

Co-sine  of  13  Degrees. 

Co-sine  of  1  2  Degrees. 

55 

P  T,  t  p  "  2"  3"  4"  5"  f,"  7"  8"  0" 

.  (  1"  2"  3"  4"  5"  6"  7"  8"  9" 

in\  1   122   334   4   5 

in{  011223344 

LOGARITHMIC    T  A  N  a  R  N  T  s. 


101 


d 

Tangent  of  76  Degrees. 

c 

Tangent  of  77  Degrees. 

s 

0"      10" 

20" 

30" 

40" 

50" 

ij 

0" 

10"   20" 

30" 

40"  (50^ 

o 

10.  603229^319 

34o8 

3498 

3588 

3678 

59 

0 

10.636636 

6732 

6828 

6924 

7020 

7116 

59 

I 

3767 

3857 

3947 

4o37 

4127 

4217 

58 

I 

7213 

73oq 

74o5 

75oi 

7597 

7694 

58 

2 

43o6 

4396 

4486 

4576 

4666 

4756 

57 

2 

779° 

7886(7983 

8o79 

8i75 

8272 

57 

3 

4846 

4936 

5o26 

5n6 

52o6 

5296 

56 

3 

8368 

8465 

856i 

8657 

8754 

885o 

56 

4 

5386 

5477 

5567 

5657 

5747 

5837 

55 

4 

8947 

9043 

9i4o 

9237 

9333 

943o 

55 

5 

5927 

6017 

6108 

6i98 

6288 

6378 

54 

5 

9526 

9623 

9720 

98i6 

99i3 

.  .10 

54 

6 

6469 

6559 

6649 

674o 

683o 

6920 

53 

6 

10.640107 

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o3oo 

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53 

7 

7011 

7101 

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7282 

7372 

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52 

7 

0687 

0784 

0881 

0978 

1075 

1172 

52 

8 

7553 

7644 

7734 

7825 

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8006 

5i 

8 

I269 

i366 

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1657 

1754 

5i 

9 

8097 

8187 

8278 

8368 

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855o 

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9 

i85i 

1948 

2046 

2143 

2240 

2337 

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10 

10.  6o864o 

873i 

8822 

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49 

10 

10.642434 

253i 

2629 

2726 

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49 

IT 

9i85 

92?6 

9367 

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9548 

9639 

48 

ii 

3oi8 

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34o7 

35o5 

48 

12 

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47 

12 

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3797 

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10.  610276 

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1186 

1278 

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1  5 

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44 

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536o 

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5555 

5653 

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44 

16 

1916 

2008 

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16 

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6339 

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18 

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18 

7124 

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7320 

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3562 

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20 

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39 

20 

io.6483o3 

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39 

21 

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38 

21 

8894 

8993 

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9288 

9387 

38 

22 

52i4 

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5398 

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5582 

5674 

37 

22 

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9880 

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23 

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36 

23 

io.65oo78 

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24 

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6687 

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35 

24 

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0770 

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1067 

1166 

35 

25 

6871 

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34 

25 

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i562 

1661 

1760 

34 

26 

7425 

75i8 

7610 

7702 

7795 

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33 

26 

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1958 

2o58 

2157 

2256 

2355 

33 

27 

7980 

8072 

8i64 

8257 

8349 

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32 

27 

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2554 

2653 

2752 

2852 

295l 

32 

28 

8534 

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28 

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29 

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29 

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10.619646 

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29 

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10.654245 

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22 

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21 

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9152 

9253 

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21 

39 

468i 

4775 

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20 

39 

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9757 

9857 

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10.625244 

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5807 

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5995 

6o89 

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18 

4i 

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1170 

1271 

1372 

10 

42 

6371 

6465 

6559 

6653 

6747 

684i 

17 

42 

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1676 

1777 

1878 

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43 

6936 

7o3o 

7124 

7218 

73l2 

7407 

16 

43 

2081 

2182 

22«3 

2385 

2486 

2587 

16 

44 

75oi 

7595 

7689 

7784 

7878 

7972 

i5 

44 

2689 

2790 

2892 

2993 

3o95 

3i96 

i5 

45 

8067 

8161 

8256 

835o 

8444 

8539 

i4 

45 

3298 

3399 

35oi 

36o2 

37o4 

38o6 

i4 

46 

8633 

8728 

8822 

89i7 

9on 

9106 

i3 

46 

39°7 

4oo9 

4in 

4212 

43i4 

44i6 

i3 

47 

9201 

9295 

939o 

9484 

9579 

9674 

12 

47 

45i8 

4620 

4721 

4823 

4925 

5027 

12 

43 

9768 

9863 

9958 

..53 

•  i47 

.242 

I  I 

48 

5i29 

523i 

5333 

5435 

5537 

5639 

II 

49 

io.63o337 

o432 

0527 

0622 

0716 

08  1  1 

IO 

49 

574i 

5843 

5945 

6047 

6i49 

6252 

10 

5o 

10.630906 

1001 

1096 

n9i 

1286 

i38i 

9 

5o 

10.666354 

6456 

6558 

6660 

6763 

6865 

9 

5i 

1476 

i57i 

1666 

i76i 

i857 

I952 

8 

5i 

6967 

7070 

7172 

7274 

7377 

7479 

8 

62 

2047 

2l42 

2237 

2332 

2428 

2523 

7 

52 

7582 

7684 

7787 

7889 

7992 

8094 

7 

53 

2618 

2713 

28o9 

2904 

2999 

3o95 

6 

53 

8197 

8299 

8402 

85o5 

8607 

8710 

6 

54 

Sigo 

3285 

338i 

3476 

3572 

3667 

5 

54 

88i3 

89i6 

9oi8 

9I2I 

9224 

9327 

5 

55 

3763 

3858 

3954 

4o49 

4i45 

4240 

4 

55 

943o 

9532 

9635 

9738 

984i 

9944 

4 

56 

4336 

4432 

4527 

4623 

4718 

48i4 

3 

56 

10.670047 

oi5o 

0253 

o356 

o459 

o562 

3 

57 

4910 

5oo6 

5ioi 

5i97 

5293 

5389 

2 

57 

0666 

O769 

0872 

o975 

1078 

1181 

2 

58 

5485 

558o 

5676 

5772 

5868 

5964 

I 

58 

1285 

1  388 

i49i 

i595 

i698 

1801 

I 

59 

6060 

6i56 

6252 

6348 

6444 

654o 

0 

59 

1905 

2008 

2112 

22l5 

23i8 

2422 

O 

60" 

50" 

40" 

30" 

20" 

10" 

c 

60"     50"   40"   30"   20"  |  10" 

a 

Co-tangent  of  13  Degrees. 

3 

Co-  tangent  of  12  Degrees. 

£a 

3 

P  ParfJ  *"  2"  3</  4"  5"  6//  7//  8"  9" 
irl$  9  19  28  37  46  56  65  74  83 

.  (  I"  2"  3"  4"  5"  6"  7"  8"  9" 
I  10  20  30  40  50  GO  70  80  90 

102 


LOGARITHMIC    SINES. 


.5 

Sine  of  78  Degrees. 

a 

Sine  of  79  Degrees. 

i  —  — 

5 

0" 

10"  f  20"  |  30"   40"   50" 

§ 

Ov      10" 

20"  f  30" 

40' 

50" 

o 

I 

2 

9.990404 
o43  1 

o458 

o4o9]o4i3  o4i8 
o436Jo44o  o445 
0462  0467  0471 

0422 
o449 
0476 

0427 
o454 
o48o 

59 

58 
5? 

o 
I 

2 

9.99i947 
1971 
I996 

,961 
i975 

2OOO 

i955 
i979 

200/i 

i959 

i983 
2008 

i963 
i987 

2OI2 

i967 

I992 

2016 

59 
58 
57 

0 

w 

4 

o485 
o5u 

0489  0494  0498  o5o3 
o5i6Jo52oio525  0529 

0507 
o534 

56 
55 

4 

2O20 

204/ 

2O24 
2O49 

2O28 

2o53 

2032 

2o57 

2036 
2061 

2040 

2o65 

56 

55 

K 

o538 

o543  o547  o552  o556 

o56o 

54 

5 

2o69 

2O73 

2077 

2081 

2o85 

2o89 

54 

6 

o565 

o569'o574o578o583 

o587 

53 

6 

2O9S 

2097 

2IOI 

2IO5 

2IO9 

2113 

53 

7 

0591 

0596  0600  o6o5 

0609 

0614 

52 

7 

2118 

2122 

3126 

2i3o 

2l34 

2i38 

52 

8 

0618 

0622  0627  o63i 

o636 

o64o 

5i 

8 

2l42 

2146 

2i5o 

2i54 

2i58 

2162 

5i 

9 

o645 

0649  o653  o658 

0662 

0667 

5o 

9 

2166 

2I7O 

2I74 

2I78 

2182 

2186 

5o 

10 

9.990671 

0670  0680  0684 

0689 

0693 

49 

10 

9.992190 

2I94 

2I98 

22O2 

2206 

2210 

49 

:i 

0697 

0702  0706  0711 

0715 

0719 

48 

ii 

22l4 

22l8 

2222 

2226 

2230 

2234 

48 

12 

0724 

0728^7330737 

0741 

o746 

4? 

12 

2239 

2243 

2247 

225l 

2255 

2259 

4? 

i3 

0750 

o755|o759 

0763 

0768 

0772 

46 

i3 

2263 

2267 

2271 

2275 

2279 

2283 

46 

i4 

0777 

0781  o785 

0790 

0794 

0798 

45 

i4 

2287 

229I 

2295 

2299 

2303 

23o7 

45 

i5 

o8o3 

08070812 

0816 

0820 

0825 

44 

i5 

23ll 

23l5 

23l9 

!i323 

2327 

233i 

44 

16 

0829 

o833o838 

0842 

o847 

o85i 

43 

16 

2335 

2339 

2343 

2347 

235i 

2355 

43 

*7 

18 

o855 

0882 

0860  o864  0868 
0886^8900895 

0873 
0899 

0877 
0903 

42 

4i 

»7 

18 

2359 

2382 

2363 
2386 

2366 
239o 

2370 
2394 

2374 
2398 

2378 
2402 

42 

4i 

*9 

0908 

0912  0916 

0921 

0925 

0929 

4o 

*9 

2406 

24  10 

24i4 

24i3 

2422 

2426 

4o 

20 

9.990934 

0938  0942 

0947 

o95  1 

o955 

39 

20 

9.992430 

2434 

2438 

2442 

2446 

245o 

39 

21 

0960 

0964^969 

0973 

0977 

0982 

38 

21 

2454 

2458 

2462 

2466 

2470 

24?4 

38 

22 

0986 

09900995 

0999 

ioo3 

1008 

37 

22 

2478 

2482 

2485 

2489 

2493 

2497 

37 

23 

1012 

10161021 

1025 

1029 

io33 

36 

23 

25oi 

25o5 

25o9 

25i3 

2517 

2521 

36 

24 

io38 

1042  io46 

io5i 

io55 

loSg 

35 

24 

2525 

2529 

2533 

2537 

254i 

2545 

35 

25 

io64 

1068  1072 

1077 

1081 

lo85 

34 

25 

2549 

2553 

2556 

256o 

2564 

2568 

34 

26 

1090 

1094  1098 

uo3 

1107 

mi 

33 

26 

2572 

2576 

258o 

2584 

2588 

2592 

33 

27 

iu5 

1120  1124 

1128 

n33 

1137 

32 

27 

2596 

2600 

2604 

2607 

2611 

26i5 

32 

28 

n4i 

u46  ii5o 

u54 

n58 

n63 

3i 

28 

2619 

2623 

2627 

263i 

2635 

2639 

3i 

29 

1167 

1171  1176 

1180 

1184 

1188 

3o 

29 

2643 

2647 

265i 

2664 

2658 

2662 

3o 

3o 

9.991193 

1197  1201 

1206 

I2IO 

1214 

29 

3o 

9.992666 

2670 

2674 

2678 

2682 

2686 

29 

3i 

1218 

I223|l227 

I23l 

1235 

I24O 

28 

3i 

2690 

2693 

2697 

2701 

2705 

2709 

28 

32 

1244 

I248!i253 

1257 

1261 

1265 

27 

32 

2?I3 

27I7 

272I 

2725 

2728 

2732 

27 

33 

1270 

1274,1278 

1282 

1287 

1291 

26 

33 

2736 

2740 

2744 

2748 

2752 

2756 

20 

34 

1295 

1299  1  3o4 

i3o8 

I3l2 

i3i6 

25 

34 

2759 

2763 

2767 

2771 

2775 

2779 

25 

35 

1321 

i325  1329 

i333 

i338 

1  342 

24 

35 

2783 

2787 

2790 

2794 

2798 

2802 

24 

36 

1  346 

i35o|i355 

i359 

i363 

i36? 

23 

36 

2806 

2810 

28l4 

2818 

2821 

2825 

23 

i  37 

1372 

i376'i38o 

1  384 

i389 

1393 

22 

3? 

2829 

2833 

2837 

2841 

2845 

2848 

22 

38 

i397 

i4oi  i4o6 

i4io 

i4i4  i4i8 

21 

38 

2852 

2856 

2860 

2864 

2868 

2871 

21 

39 

1422 

1427  i43i 

i435 

i439  i444 

2O 

39 

2875 

2879 

2883 

2887 

289i 

2895 

2O 

4o 

9.991448 

i452  i456 

i46o 

i465  1469 

*9 

4o 

9.992898 

2902 

29o6 

2910 

29i4 

29l8 

r9 

4i 

i473 

1477  1482 

i486 

1490  1494 

18 

4i 

2921 

2925 

2929 

2933 

2937 

294l 

18 

42 

1498 

i5o3  1507 

i5u 

i5i5  iSig 

J7 

42 

2944 

2948 

2952 

2956 

296o 

2963 

J7 

43 

i524 

i528  i532 

i536 

i54oi545 

16 

43 

2967 

297I 

2975 

2979 

2983 

2986 

16 

44 

1  549 

i553li557 

i56i 

i566  1570 

i5 

44 

2990 

2994 

2998 

3OO2 

3oo5 

3oo9 

i5 

45 

i574 

i578'i582 

:586 

1691  1595 

i4 

45 

3oi3 

3oi7 

3O2I 

3O24 

3028 

3o32 

i4 

46 

i599 

i6o3  1607 

1612 

1616  1620 

i3 

46 

3o36 

3o4o 

3o43 

3o47 

3o5i 

3o55 

i3 

4? 

1624 

i628'i632 

i637 

i64iji645 

12 

47 

3o59 

3o62 

3o66 

3o7o 

3o74 

3o78 

12 

48 

1649 

i653:i657 

1662 

1666  1670 

II 

48 

3o8i 

3o85 

3o89 

3o93 

3o97 

3ioo 

II 

49 

1674 

1678  1682 

1687 

1691  1695 

IO 

49 

3io4 

3io8 

3iu 

3n5 

3n9 

3i23 

IO 

5o 

9.991699 

1703  1707 

1712 

1716  1720 

9 

5o 

9.993i27 

3i3i 

3i34 

3i38 

3l42 

3i46 

9 

5i 

1724 

1728,1732 

i736 

1741  1745 

8 

5i 

3i49 

3i53 

3!57 

3i6i 

3i65 

3i68 

8 

52 

1749 

1753  1757 

1761 

1765  1770 

7 

52 

3l72 

3i76 

3i8o 

3i83 

3i87 

3i9i 

7 

53 

1774 

I778ii782 

1786 

1790  i794 

6 

53 

3195 

3i98 

3202 

3206 

3210 

32i3 

6 

54 

1799 

1803,1807 

1811 

i8i5  1819 

5 

54 

32I7 

3221 

32a5 

3228 

3232 

3236 

5 

55 

i823 

1827  i832 

i836 

i84o  1  844 

4 

55 

324o 

3243 

3247  325i 

3255 

3258 

4 

56 

1  848 

i852;i856 

1860 

i865  1869 

3 

56 

3262 

8266  327o  3273 

3277 

328i 

3 

5? 

i873 

1877  1881 

i885 

1889  1893 

2 

57 

3284 

3288  3s92  3296 

3299  33o3 

2 

58 

1897 

1901  1906 

1910 

1914  1918 

I 

58 

33o7  33n  33i4  33i8  33s?  3325 

I 

59 

1922 

1926  193011934 

1938  1942 

O 

59 

3329>3333  3337  334o  3344  3348 

O 

60" 

50"  1  40"   30"   20"   10" 

g 

60"    j  50"   40"   30"   'JO"  '  10"   ^ 

Co-sine  of  1  1  Decrees. 

2, 

Co-sine  of  1  0  Degrees.     2 

p  p  .  $  1"  2"  3"  4"  5"  6"  7"  8"  9" 
irl$  0   1   1   2   2   3   3   3   4 

<  1"  2"  3"  4"  5"  C"  7"  8"  9" 
P.  Tart  J  o   i   !   o   o   o   3   ;j   4 

LOGARITHMIC    TANGENTS. 


103 


A 

Tangent  of  78  Degrees. 

a 

Tangent  of  79  Degrees. 

S 

0" 

10" 

20" 

30" 

40" 

50" 

§ 

0" 

10" 

20" 

30" 

40" 

50" 

o 

10.672525 

2620, 

2733 

2836 

294o 

3o43 

r 

o 

10.711348 

i46o 

i573 

i685 

i798 

I9IO 

59 

I 

3i47 

325i 

3354 

3458 

3562 

3666 

i 

2023 

2i35 

2248 

236i 

2473 

2586 

58 

2 

3769 

3873 

3977 

4o8i 

4i85 

4289 

57 

2 

2699 

2811 

2924 

3o37 

3i5o 

3263 

57 

3 

4393 

4497 

46oi 

47o5 

48o9 

49*3 

56 

3 

3376 

3488 

36oi 

37i4 

3827 

394o 

56 

4 

5017 

5l2I 

5225 

5329 

5433 

5537 

55 

4 

4o53 

4i67 

4280 

4393 

45o6 

46i9 

55 

5 

5642 

5746 

585o 

5954 

6o59 

6i63 

54 

5 

4732 

4846 

4959 

5072 

5i85 

5299 

54 

6 

6267 

6372 

6476 

658o 

6685 

6789 

53 

6 

54i2 

5526 

5639 

5752 

5866 

5979 

53 

7 

6894 

6998 

7io3 

7207 

7312 

7417 

52 

7 

6o93 

62O7 

6320 

6434 

6547 

6661 

52 

8 

7521 

7626 

773i 

7835 

7940 

8o45 

5i 

8 

6775 

6889 

7OO2 

7n6 

723o 

7344 

5i 

9 

8i49 

8254 

8359 

8464 

8569 

8674 

5o 

9 

7458 

7572 

7686 

7799 

79i3 

8027 

5o 

10 

8778 

8883 

8988 

9o93 

9i98 

93o3 

49 

10 

8142 

8256 

837o 

8484 

8598 

8712 

49 

ii 

94o8 

95i3 

96i8 

9723 

9829 

9934 

48 

ii 

8826 

894i 

9o55 

9i69 

9283 

9398 

48 

12 

io.68oo39 

oi44 

O249 

o355 

o46o 

o565 

47 

12 

95i2 

9627 

974i 

9856 

9970 

..85 

47 

i3 

0670 

o776 

0881 

o987 

I092 

n97 

46 

i3 

IO.72OI99 

o3i4 

0428 

o543 

o658 

0772 

46 

14 

i3o3 

i4o8 

i5i4 

i6i9 

1725 

i83o 

45 

i4 

0887 

1002 

1116 

I23l 

1  346 

i46i 

45 

r5 

I936 

2042 

2147 

2253 

2359 

2464 

44 

i5 

i576 

i69i 

1806 

1921 

2o36 

2l5l 

44 

16 

2570 

2676 

2782 

2887 

2993 

3o99 

43 

16 

2266 

238i 

2496 

2611 

2726 

2841 

43 

17 

32o5 

33ii 

3417 

3523 

3629 

3735 

42 

17 

2957 

3o72 

3i87 

3302 

34i8 

3533 

42 

18 

384i 

3947 

4o53 

4i59 

4265 

437i 

4i 

1.8 

3649 

3764 

3995 

4no 

4226 

4i 

'9 

4477 

4584 

469o 

4796 

49O2 

5oo9 

4o 

i9 

4342 

4457 

4573 

4688 

48o4 

4920 

4o 

20 

5n5 

5221 

5328 

5434 

5540 

5647 

39 

20 

5o36 

5i5i 

5267 

5383 

5499 

56i5 

39 

21 

5753 

586o 

5966 

6o73 

6179 

6286 

38 

21 

573i 

5847 

5o63 

6o79 

6i95 

63n 

38 

22 

6392 

6499 

6606 

67I2 

68i9 

60,26 

37 

22 

6427 

6543 

6l55o 

6775 

689i 

7008 

37 

23 

7032 

7i39 

7246 

7353 

7460 

7567 

36 

23 

7I24 

724o 

7356 

7473 

7589 

7706 

36 

24 

7673 

778o 

7887 

7994 

8101 

8208 

35 

24 

•7822 

7938 

8o55 

8i7i 

8288 

84o5 

35 

25 

83i5 

8422 

8529 

8636 

8744 

885i 

34 

25 

852i 

8638 

8754 

887i 

8988 

9io5 

34 

26 

895.8 

9o65 

9I72 

928o 

9387 

9494 

33 

26 

922I 

9338 

9455 

9572 

9689 

98o6 

33 

27 

96oi 

9709 

98i6 

9924 

.138 

32 

27 

9923 

..4o 

;l57 

.274 

.39i 

.5o8 

32 

28 

10.690246 

o353 

o46i 

o568 

0676 

0784 

3i 

28 

io.73o625 

o742 

0860 

0977 

io94 

I2II 

3i 

29 

o89i 

°999 

1107 

I2l4 

1322 

i43o 

3o 

29 

i329 

1  446 

i563 

1681 

i798 

i9i6 

3o 

3o 

i537 

1  645 

i753 

2861 

i969 

2077 

29 

3o 

2o33 

2l5l 

2268 

2386 

25o3 

2621 

29 

3ii     2i84 

2292 

2400 

25o8 

2616 

2724 

28 

3i 

2739 

2856 

2974 

3o92 

32IO 

3327 

28 

3aj     2832 

294i 

3o49 

3i57 

3265 

3373 

27 

32 

3445 

3563 

368i 

3799 

39i7 

4o35 

27 

33 

348  1 

359o 

3698 

38o6 

39i4 

4023 

26 

33 

4i53 

427I 

4389 

45o7 

4625 

4744 

26 

34 

4i3i 

42  39 

4348 

4456 

4565 

4673 

25 

34 

4862 

498o 

5o98 

5217 

5335 

5453 

25 

35 

4782 

489o 

4999 

5io7 

52i6 

5325 

24 

35 

5572 

569o 

58o9 

5927 

6o46 

6i64 

24 

36 

5433 

5542 

565i 

5759 

5868 

5977 

23 

36 

6283 

64oi 

6520 

6639 

6757 

6876 

23 

37 

6086 

6i95 

63o3 

6412 

652i 

663o 

22 

37 

6995 

7n3 

7232 

735i 

7470 

7589 

22 

38 

6739 

6848 

6957 

7o66 

7175 

7284 

21 

38 

77o8 

7827 

7946 

8o65 

8184 

83o3 

21 

39 

7393 

75o3 

7612 

772i 

783o 

7939 

20 

39 

8422 

854i 

8660 

8780 

8899 

9oi8 

2O 

4o 

8o49 

8i58 

8267 

8376 

8486 

8595 

I9 

4o 

9i37 

9257 

9376 

9496 

96i5 

9734 

I9 

4i 

87o5 

88i4 

8924 

9o33 

9i43 

9252 

18 

4i 

9854 

9973 

..93 

.213 

.332 

.452 

18 

42 

9362 

947i 

958i 

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o 

80" 

50" 

40" 

30" 

20" 

10" 

g 

bO" 

50" 

40" 

30" 

20"   10" 

a 

Co-tangent  of  1  1  Degrees. 

Co-tangent  of  10  Degrees. 

i 

P  Part$  l"  ~"  3"  4"  5"  6//  7"  8"  9// 

,  (  1"  2"  3"  4"  5"  6"  7"  S"  9" 

I  11  22  32  43  54  65  75  86  97 

irt\  12  23  35  47  59  70  82  94  10G 

104: 


LOGARITHMIC    SINKS. 


1 

Sine  of  80  Degrees. 

c 

Sine  of  81  Degrees 

SI 

0" 

10" 

20'' 

30"' 

40" 

50" 

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0" 

10" 

20" 

30" 

40" 

50" 

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0 

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0 

60" 

50" 

40" 

30"   20" 

10" 

ri 

60-''     50"   40"  |  30"  :  20"  i  10"  j 

Co-sine  of  9  Degrees. 

g 

Co-sine  of  8  Degrees.      ^ 

f>  1"  2"  3"  4"  5"  6"  7"  8"  9" 
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Pp,rf$  1"  2"  3"  4"  5"  G'1  7"  8"  i,' 

in}  o  1  i  i  2  2  2  n  3 

L  O  G  A  R  I  T  II  M  f  C      T  A  N  G  E  X  T  S. 


105 


|j    Tangent  of  80  Degrees. 

c  1    Tangent  of  8  1  Degrees. 

^ 

0" 

10"  |  20"   30" 

40" 

|  50" 

2 

0"      10" 

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30" 

40"   50"  ] 

0 

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4667 

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4961 

19 

4i 

5oi  i 

5i42 

5274 

54o6 

5538 

5670 

18 

4i 

5i68 

5255 

54o2 

555o 

5697 

5844 

18 

42 

58o2 

5934 

6066 

6i98 

633o6463 

17 

42 

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6434 

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43 

6595 

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7468  7616 

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44 

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7764 

7912 

8060 

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45 

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45 

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48 

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48 

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1  648 

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49 

2225 

2375 

2524 

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2973 

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9 

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3 

52 

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7 

52 

4923 

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5224  5374 

5525(5675 

7 

53 

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4734 

4869 

5oo4 

5138,5273 

6 

53 

5826 

5977 

6127 

6278 

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6 

54 

54o8 

5543 

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5947  6082 

5 

54 

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5 

55 

6218 

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4 

55 

7637 

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4 

56 

7029 

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3 

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3 

57 

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2 

57 

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2 

58 

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53 

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59 

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9879  ..i5 

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59 

1282 

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2045 

O 

.60"      50"  |  40" 

30"   20" 

10" 

. 

60" 

50" 

40" 

30" 

?0"  I  10" 

Co-tangent  of  9  Degrees. 

1 

Co-tangent  of  8  Degrees. 

.3 

P  PartJ  l"  2"  3"  4"  5"  6"  7"  8"   9" 
IIIJ  13  26  39  52  65  78  91  103  116 

p  p   (  1"  2"  3"  4"  5"  6"  7"  8"  9" 
1  $  ,14  29  43  58  72  8G  101  115  130 

10G 


L  o  G  A  K  i  r  ii  M  i  c    SINES. 


1 

Sine  of  82  Degrees. 

a 

Sine  of  83  Degrees. 

51 

0" 

10" 

20" 

30" 

40" 

50" 

i 

0" 

10" 

20" 

30" 

40" 

50" 

0 

9.996753 

5756 

5759 

5762 

5765 

5768J59 

o 

9.99675i!6753 

6756 

6758 

676i 

6764 

59 

i 

5771 

5773 

5776 

5779 

5782 

5785 

58 

I 

6766 

6769 

677i 

6774 

6777 

6779 

58 

2 

5788 

579i 

5794 

5797 

58oo 

58o3 

57 

2 

6782 

6784 

6787 

6789 

6792 

6795 

57 

3 

58o6 

5809 

58i2 

58i5 

58i8582i 

56 

3 

6797 

6800 

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56 

4    5823 

5826 

5829 

5832 

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55 

4 

6812 

68i5 

6818 

6820 

6823 

6825 

55 

5    584i 

5844 

5847 

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5853 

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54 

5 

6828 

683o 

6833 

6835 

6838 

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54 

6 

5859 

5862 

5864 

5867 

587o 

587353 

6 

6843 

6846 

6848 

685i 

6853 

6856 

53 

7 

5876 

5879 

5882 

5885 

5888 

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7 

6858 

6861 

6863 

6866 

6869 

687i 

D2 

8 

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5899 

5902 

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59o8 

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8 

6874 

6876 

6879 

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9 

59n 

5qi4 

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5923 

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689i 

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109.995928 

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49 

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48 

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6919 

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12 

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5972 

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47 

12 

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6937 

6939 

6942 

6944 

6947 

47 

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5980 

5983 

5986 

5989 

5992 

5995 

46 

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6949 

6952 

6954 

6957 

6959 

6962 

46 

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6001 

6oo3  6006 

6oo9 

6012 

45 

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6964 

6967 

6969 

6972 

6974 

6977 

45 

i5 

6oi5 

6018 

6021 

6023 

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44 

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6979 

6982 

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44 

16 

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16 

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18 
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4o 

20 

9.996100 

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6106  6  1  o9 

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39 

20 

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21 

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6i3i 

38 

21 

7068 

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7o76 

7o78 

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22 

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6i37 

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37 

22 

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7o85 

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37 

23 

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36 

23 

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25 

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34 

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29 

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28 

32 

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27 

32 

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27 

33 

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24 
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22 

38 

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39 

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55 

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4 

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3 

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3 

57 

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2 

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58 

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&y   \  50" 

40" 

30"   20"  |  10" 

n 

60"     50"   40"   30"  i  20"   10"  i  rf 

Co-sine  of  7 

Degrees. 

i 

Co-sine  of  6  Degrees. 

C  1"  2"  3"  4" 
IM'artJ  0   !   !   j 

5"  6"  7"  8"  9" 
12222 

p  p  .  $  1"  2"  3"  4"  5"  6"  7"  8"  9" 
irt\  0   0   1   1   1   1   2   2   2 

LOGARITHMIC    TANGENTS. 


107 


J 

Tangent  of  82  Degrees. 

.s 

Tangent  of  83  Degrees. 

s 

0" 

10" 

20" 

30" 

40"' 

50" 

s 

0"     |  10"   20" 

30" 

40" 

50" 

o 

30.80219,7 

235o 

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2962 

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55 

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5 

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7 

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8 

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48 

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18 

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42 

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2608 

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3277 

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42 

7027 

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3 

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67 

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57 

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59566i57 
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I 

0 

60" 

50" 

40" 

30" 

20" 

10" 

, 

60"    |  50"   40"  |  30" 

20" 

10" 

s  i 

Co-tangent  of  7  Degrees. 

S 

Co-tangent  of  6  Degrees. 

9 

p  p   C  1"  2"  3"  4"  5"  6"  7"  8"  9" 
1  I  16  33  49  65  81  98  114  130  146 

p  p  .(  1"  2"  3"  4"  5"  6"  7"  8"  9'' 
I  19  37  50  75  94  112  131  IfC  1U8 

108 


LOGARITHMIC    SINES. 


.s 

Sine  of  34  Degrees. 

d 

Sine  of  85  Degrees. 

3 

0"    i  10"  |  20"   30"   40" 

50" 

& 

0"    |  10" 

20"   30" 

40-' 

50" 

0 

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2 

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2 

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57 

3 

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7 

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10 

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ii 

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48 

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8464 

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12 

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778o 

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12 

8474 

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8479 

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47 

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46 

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46 

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44 

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16 

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18 

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19 

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788i 

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20 

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21 

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38 

21 

8568 

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22 
23 

7897 
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7900  7902  7904 
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36 

22 
23 

8578 
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24 

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28 

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29 

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3i 

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8010  8012  8oi4  8016  8018 

28 

3i 

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28 

32 

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8022  8024  8026  8028  8o3o 

27 

32 

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36 

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8070  8072  8074 

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23 

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17 

4i 

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8766 
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44 

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47 

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8199  8201  8203  8205,8207 

12 

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8824 

8826 

8827 

8829 

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12 

48 

8209 

8211  82i382i5 

82178218 

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48 

8832 

8834 

8835 

8837 

8838 

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49 

8220 

8222  8224  8226 

82288230 

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49 

884i 

8843 

8844 

8846 

8848 

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9.998232 
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823482368238 
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8239824i 
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5o 
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8852 
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8854 
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8855 
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8857 
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52 

8255 

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7 

52 

8869 

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8872 

8873 

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53 

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8268  8270  8272 

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53 

8878 

8880 

8881 

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8884 

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54 

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5 

54 

8887 

8889 

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5 

55 

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8290  8292  8294 

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4 

55 

8896 

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4 

56 

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3 

56 

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3 

57 

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2 

57 

89i4 

89i6  89i7  89i9 

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2 

58 
59 

8322 

8333 

8324  8326  8328  8329  833i 
8335  83378339834i  8342 

I 
O 

58 
59 

8923  8925  8926  892-7  8929  893o 
8932:8933  8935  8936  8938  8939 

0 

60" 

50"  j  40"   30"   20"   10"  |  ^ 

60"    ;  50"   40"   30"   20"   10"   ^ 

Co-sine  of  5  Degrees.     j  § 

Co-sine  of  4  Degrees.     j  § 

<  i"  2"  3"  4"  5"  6"  7"  8"  9" 
all  001111122 

„  „  .  <  1"  2"  3"  4"  5"  6"  7"  8"  9" 
irt\  0   0   0   1   1   1   1   1   1 

o  G  A  it  i  T  ii  M  i  c    TANGENTS. 


109 


g 

Tangent  of  84  Degrees. 

_s- 

Tangent  of  85  Degrees.     ! 

s 

0" 

10" 

20" 

30" 

40" 

50" 

S 

0"     |  10"  j  20" 

30" 

40"  |  50" 

0 

10.978380 

8582 

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8988 

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n.o58o48 

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3 

2435 

2680 

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4 

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27l3 

2994 

3275 

3556 

18 

42 

2606 

2835 

3o64 

3293 

3522 

3752 

17 

42 

3838 

4n9 

44oi 

4683 

4966 

5248 

17 

43 

398i 

4211 

444  1 

467i 

49oi 

5i3i 

16 

43 

553i 

58i3 

6o96 

638o 

6663 

6946 

16 

44 

536i 

5592 

5822 

6o53 

6284 

65i4 

i5 

44 

723o 

75i4 

7798 

8082 

8367 

865i 

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45 

6745 

6977 

7208 

7439 

7671 

7902 

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45 

8936 

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9792 

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i-4 

46 

8i34 

8366 

8598 

883o 

9062 

9295 

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46 

n.i3o649 

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1221 

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1794 

2081 

13 

47 

9527 

976o 

9992 

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.458 

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13 

47 

2368 

2656 

2943 

323i 

35i8 

38o6 

12 

48 

ii  .  o4o925 

n58 

i39i 

1625 

i859 

2O92 

II 

48 

4094 

4383 

467i 

4960 

5249 

5538 

II 

49 

2326 

256i 

2795 

3O29 

3264 

3498 

10 

49 

5827 

6.117 

64o7 

6697 

6987 

7277 

IO 

5o 

3733 

3968 

4203 

4438 

4673 

49o8 

9 

5o 

7567 

7858 

8i49 

844o 

873i 

9 

5i 

5i44 

5379 

56i5 

585i 

6087 

6323 

8 

5i 

96o6 

9898 

.  190 

.483 

9775 

8 

52 

6559 

6795 

7o32 

7268 

75o5 

7742 

7 

52 

n.i4io68 

i36i 

i654 

1947 

2241 

2535 

7 

53 

7979 

8216 

8453 

8691 

8928 

9i66 

6 

53 

2829 

3i23 

3417 

3712 

4007 

43oi 

6 

54 

94o3 

964i 

9879 

.117 

.356 

•  594 

5 

54 

4597 

4892 

5i87 

5483 

5779 

6o75 

5 

55 

ii.o5c832 

1071 

i3io 

1  549 

1788 

2027 

4 

55 

6372 

6668 

6965 

7262 

7559 

7856 

4 

56 

2266 

25o6 

2745 

2985 

3225 

3465 

3 

56 

8i54 

8452 

875o 

9o48 

9346 

9645 

3 

57 

37o5 

3945 

4i85 

4426 

4666 

4907 

2 

57 

9943 

.242 

.542 

.84  1 

i  i4i 

i44o 

2 

58 

5i48 

5389 

563o 

5871 

6112 

6354 

I 

58 

2o4l 

234l 

2642 

2942 

3243 

I 

5  9 

6596 

6837 

^o_79 

732I 

7563 

7806 

0 

59 

3545 

3846 

4i48 

4449 

4752  5o54 

O 

60"      50"  |  40" 

30"  |  20" 

10" 

d 

|    60"      50" 

40"   30"   20"   10'7 

S* 

Co-tangent  of  5  Degrees. 

.3 

Co-tangent  of  4  Degrees. 

. 

P  PartM"2"  3//  4//  5"  G"  7"  8//   9" 

„  p  .  5  .1"  2"  3"  4",  5"  6"  7"  &'  9" 

\  22  44  66  88  110  132  154  177  199 

Lrt)  27  54  81  108  135  162  188  215  242 

110 


•LOGARITHMIC    SINES. 


I 

Sine  of  86  Degrees. 

a 

Sine  of  87  Degrees. 

m 

0" 

10" 

20"   30"   40"   50" 

i 

0" 

10" 

20" 

30" 

4,. 

50" 

o 

9.99894i 

8942 

8944  8945  8947 

8948 

59 

o 

9.999404 

94o6 

94o7 

94o8 

94o9 

94io 

59 

1 

895o 

895i 

895389548955 

8957 

58 

I 

94n 

94l2 

94i3 

94i4 

94:5 

9416 

58 

2 

8958 

8960 

8961  8963 

8964 

8966 

57 

f 

94i8 

94i9 

9420 

94si 

9422 

9423 

57 

3 

8967 

8969 

8970  8971 

8973 

8974 

56 

: 

9424 

9425 

0426 

9427 

9428 

943o 

56 

4 

8976 

8977 

8979  8980 

8982 

8983 

55 

L 

943i 

9432 

9433 

9434 

9435 

9436 

55 

5 

8984 

8986 

8987  8989 

8990 

8992 

54 

t 

9437 

9438 

9439 

944o 

944i 

9442 

54 

6 

8993 

8995 

899618997 

8999 

9000 

53 

6 

9443 

9445 

9446 

9447 

9448 

9449 

53 

7 

9002 

9003 

9005  9000 

9007 

9009 

52 

7 

945o 

945i 

9452 

9453 

9454 

9455  52 

8 

9010 

9012  9013 

9Ol5 

9016 

9017 

5i 

8 

9456 

9457 

9458 

9459 

9460 

946i 

5i 

9 

9019 

9020 

9022 

9023 

9024 

9026 

5o 

9 

9463 

9464 

9465 

9466 

9467 

9468 

5o 

10 

9.999027 

9029 

9o3o 

9032 

9o33 

9034 

49 

10 

9  999469 

947o 

94?i 

9472 

9473 

9474 

49 

ii 

9036 

9o37 

9o39 

9040 

904  1 

9o43 

48 

ii 

9475 

9476 

9477 

9478 

9479 

948o 

48 

12 

904^ 

9046 

9047 

9048 

9o5o 

9o5i 

47 

12 

948i 

9482 

9483 

9484 

9485 

9486 

47 

i3 

9o52 

9o54 

9o55  9057 

9o58 

9o59 

46 

l2 

9487 

9488 

9489 

949o 

9491 

9492 

46 

1.4 

9061 

9062 

9064  9065 

9066 

9o68 

45 

i4 

9493 

9495 

9496 

9497 

9498 

9499 

45 

i5 

9069 

9071 

9072 

9073 

9o75 

9o76 

44 

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95oo 

95oi 

9502 

95o3 

95o4 

95o5 

44 

16 

9077 

9079  9080  9082 

9083 

9o84 

43 

16 

95o6 

95°7 

95o8 

95o9 

95io 

gSn 

43 

i? 

9086 

90879088 

9090 

9091 

9o92 

42 

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95l2 

95i3 

95i4 

95i5 

95i6 

95i7 

42 

18 

9094 

9o95  9097 

9098 

9099 

9IOI 

4i 

18 

95i8 

95i9 

9520 

952I 

9522 

9523 

4i 

J9 

9102 

9103  9105 

9106 

9107 

9io9 

4o 

J9 

9524 

9525 

9526 

952? 

9527 

9528 

4o 

20 

9.999110 

9111  9113 

911^3 

9n5 

9117 

39 

20 

9.999529 

953o 

953i 

9532 

9533 

9534 

39 

21 

9118 

9120  9121 

9122 

9124 

9125 

38 

21 

9535 

9536 

9537 

9538 

9539  954o 

38 

22 

9126 

9128  9129 

9i3o 

9132 

9i33 

37 

22 

954i 

9542 

9543 

9544 

9545 

9546 

37 

23 

9134 

9i36;9i37 

9i38 

9140 

9141 

36 

23 

9547 

9548 

9549 

955o 

955i 

9552 

36 

24 

9142 

9i439i45 

9i46 

9147 

9149 

35 

24 

9553 

9554 

9555 

9556 

9557 

9557 

35 

25 

9i5o 

9i5i  9i53 

9i54 

9i55 

9i57 

34 

25 

9558 

9559 

956o 

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9562 

9563 

34 

26 

9i58 

9159  9161 

9162 

9i63 

9i65 

33 

26 

9564 

9565 

9566 

9567 

9568 

9569 

33 

27 

9166 

9167  9168  9170 

9171 

9172 

32 

27 

957° 

957i 

9572 

9573 

9573 

9574 

3a 

28 

9174 

9175:91769178 

9179 

9180 

3i 

28 

9575 

9576 

9577 

9578 

9579 

958o 

3i 

29!   9181 

9183:9184 

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9187 

9188 

3o 

29 

958i 

9582 

9583 

9584 

9585 

9586 

3o 

309.999189 

91909192 

9i93 

9i94 

9196 

29 

3o 

9.999586 

9587 

9588 

9589 

959o 

959i 

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3i 

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9201 

9202 

9203 

28 

3i 

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9593 

9594 

9595 

9596 

9597 

28 

32 

9205 

9206  9207 

9208 

9210 

9211 

27 

32 

9597 

9598 

9599 

9600 

96oi 

9602 

27 

33 

9212 

9213  9215 

9216 

9217 

9219 

26 

33 

9603 

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9606 

96o6 

9607 

26 

34 

9220 

9221  9222 

9224 

9225 

9226 

25 

34 

9608 

9609 

96io 

9611 

96l2 

96i3 

26 

35 

9227 

9229  9230 

9231 

9232 

9234 

24 

35 

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9614 

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96i7 

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24 

36 

9235 

92369237 

9239 

9240 

9241 

23 

36 

9619 

9620 

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9622 

9622 

9623 

23 

37 

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9246 

9247 

9249 

22 

37 

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9625 

9626 

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9629 

22 

38 

9250 

9251  9252 

9254 

9255 

9256 

21 

38 

9629 

963o 

963i 

9632 

9633 

9634 

21 

39 

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9258  9260 

9261 

9262 

9263 

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9635 

9636 

9637 

9638 

9639 

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9.999265 

9266  9267 

9268 

9270 

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19 

4o 

9.999640 

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9643 

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18 

4i 

9645 

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9648 

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1  8 

42 

9279 

9280  9282 

9283 

9284 

9285 

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42 

965o 

965i 

9652 

9653 

9653 

9654 

J7 

43 

9287 

9288^289 

9290 

9291 

9293 

16 

43 

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9656 

9657 

9658 

9658 

9659 

16 

44 

9294 

9295  9296 

9297 

9299 

93oo 

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44 

9660 

966i 

9662 

9663 

9663 

9664 

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45 

93oi 

9302  93o3 

93o5 

93o6 

93°7 

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45 

9665 

9666 

9667 

9668 

9668 

9669 

i4 

46 

93o8 

93099310 

9312 

93i3 

93i4 

i3 

46 

9670 

9671 

9672 

9672 

9673 

9674 

i3 

47 

93i5 

93169318 

93i9 

9320 

9321 

12 

47 

9675 

0676 

9677 

9677 

9678 

9679 

12 

48 

9322 

93239325 

9326 

9327 

9328 

II 

48 

9680 

968i 

968i 

9682 

9683 

9684 

II 

49 

9329 

933i  9332 

9333 

9334 

9335 

IO 

49 

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9687 

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10 

5o 

9.  999336 

93389339 

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9 

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9.999689 

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969i 

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9 

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8 

5i 

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9696 

9697 

9697 

9698 

8 

62 

935o 

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9354 

9355 

9356 

7 

52 

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97oo 

9701 

9702 

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7 

53 

9357 

9358 

9359 

936i 

9362 

9363 

6 

53 

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9707 

6 

54 

9364 

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9366 

9367 

9369 

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5 

54 

9708 

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9711 

9711 

9712 

5 

55 

937i 

9372 

9373 

9374 

9375 

9377 

4 

55 

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97i4|97i5 

97i6 

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4 

56 

9378 

9379 

938o 

938i 

9382 

9383 

3 

56 

9717 

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9720 

972i 

3 

57 

9384 

9385 

9387 

9388 

9389  939o 

2 

57 

9722 

9723 

9723 

9724 

9725 

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2 

58 

9391 

9392 

9393 

9394 

9396  9397 

I 

58 

9726 

9727 

9728 

9729 

9729 

973o 

I 

59 

9398 

9399 

94oo 

94oi 

94o2  94o3 

O 

59 

973i 

9732 

9732 

9733 

9734 

9735 

O 

60" 

50" 

40"   30" 

20"  |  10" 

d 

CC" 

50"   40" 

30" 

20" 

10" 

a 

Co-sine  of  3  Degrees. 

•9 

& 

Co-sine  of  2  Degrees. 

1 

p  p.  <  1"  2"  3"  4"  5"  6"  7"  8"  9" 

irll  000111111 

p  p   (  1"  2"  3"  4"  5"  6"  7"  8"  9" 
'^000001111 

o  G  A  R  i  T  ii 


TANGENT  s. 


Ill 


: 

Tangent  of  86  Degrees. 

. 

Tangent  of  87  Degrees. 

• 

0" 

10" 

20" 

30" 

40" 

50" 

2 

0" 

10" 

20"   30"  |  40" 

50" 

0 

1  1.  155356 

5659 

5962 

6265 

6568 

6872 

59 

0 

ii  .280604 

1007 

i4n 

1814 

2219 

2623 

59 

I 

7175 

7479 

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8088 

8393 

8697 

58 

i 

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4245 

4652 

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58 

= 

9OO2 
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2064 

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56 

2 

3 

5466 
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6281  6689 
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7098 
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997° 

56 

4 

2679. 

2987 

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4220 

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4 

11.290382 

0794 

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5 

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54 

5 

2860 

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6 

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53 

6 

5354 

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7 

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52 

7 

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8700 

9120 

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9962 

52 

S 

ii  .  170126 

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1066 

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8 

ii.3oo383 

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2008 

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9 

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10 

ii 

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5  1  62  5478 
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49 

48 

10 

ii 

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49 

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12 

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18 

4i 

II  .393022 

3544 

4o66 

4589 

5n3 

5637 

18 

42 

9I28 

9495 

986i 

.229 

.596 

.964 

17 

42 

6161 

6687 

7213 

774o 

8267 

8795 

17 

43 

II.24l332 

1700 

2o69 

2438 

2807 

3i77 

16 

43 

9323 

9853 

.382 

.9i3 

1  444 

i976 

16 

44 

3547 

3917 

4288 

4659 

5o3o 

54oi 

i5 

44 

11.402608 

3o4i 

3575 

4no 

4645 

5i8o 

i5 

45 

5773 

6i45 

65i8 

689i 

7264 

7637 

i4 

45 

5717 

6254 

6792 

733o 

7869 

84o9 

i4 

46 

Son 

8385 

8759 

9i34 

95oo 

9884 

i3 

46 

8949 

9490 

.574 

1117 

1661 

i3 

47 

ii  .250260 

o636 

IOI2 

i389 

1766 

2i43 

12 

47 

II.4l22o5 

2751 

32963843 

439o 

4938 

12 

48 

2521 

2899 

3277 

3656 

4o34 

44i4 

I  I 

48 

5486 

6o36 

6586  7i36 

7688 

8240 

II 

49 

4793 

5I73 

5553 

5934 

63i5 

6696 

IO 

49 

8792 

9346 

9900'  .455 

1010 

1  566 

IO 

5o 

7078 

7460 

7842 

8224 

8607 

899i 

9 

5o 

11.422123 

2681 

324013799 

4358 

49i9 

9 

5i 

9374 

9758 

.142 

.527 

.9I2 

129-7 

8 

5i 

548o 

6o42 

66o5l7i68 

7733 

8298 

s 

52 

11.261683 

2069 

2455 

2842 

3229 

36i6 

7 

52 

8863 

943o 

9997!  .565 

1  1  33 

I702 

7 

53 
54 
55 
56 

4oo4 
6337 
8683 
ii  .271041 

4392 
6727 
9075 
i435 

4780 
7117 

9467 
i83o 

5i69 
75o8 
986o 

2225 

5558 

7899 
.254 
2620 

5947 

829I 

.647 
3oi6 

6 
5 

4 
3 

53 

54 
55 
56 

11.432273 
5709 
9172 
11.442664 

2843 
6284 
9752 
3248 

34i5 
6860 
.333 

3834 

39874560 

7437|8oi5 

.915.1497 
4420  5oo7 

5i34 
8593 
2080 
5595 

5 

4 
3 

$7 

34i2 

3809 

4206 

46o3 

5ooo 

5398 

2 

57 

6i83 

6773i7363 

7954|8546 

9i38 

2 

58 
59 

5796 
8i94 

6195 
8595 

6594 
8996 

6993 
9397 

7393 
9799 

7793 

.202 

I 
O 

58 
59 

9732 
ii.4533o9 

.326  .Q2I 

3908  45o8 

i5i7 
5  1  09 

2113 

5711 

27I 

63i3 

I 
0 

60" 

50"  |  40" 

30" 

20" 

10" 

S" 

60"     50"   4'W  |  30"  |  20" 

10" 

. 

Co-tangent  of  3  Degrees 

Co-tangent  of  2  Degrees. 

i 

f  -i  II  nil  •}//   AII   r  //   p//   »//   «//   Q// 

P.  Part  J  3_  G()  1Q4  13g  ]73  2Q7  ^  276  gn 

p  p.  (  1"2"  3"  4"  5"  6"  7"  8"  9" 
irt}  48  97  145  193  242  290  338  387  435 

112 


LOGARITHMIC    SINES. 


1 

Sine  of  88  Degrees. 

jj 

Sine  of  89  Degrees. 

fl 

0" 

10" 

20" 

30" 

40" 

50" 

s 

0" 

10"  |  20" 

30" 

40" 

50" 

o 

9.999735 

9736 

9737 

973*8 

9738 

9739 

59 

0 

9.999934 

9934 

9935 

9935 

9935 

9936 

59 

I 

974o 

974o 

974i 

9742 

9743 

9743 

58 

i 

9936 

9936 

9937 

9937 

9937 

9938 

58 

2 

9744 

9745 

9746 

9746 

9747 

9748 

57 

2 

9938 

9939 

9939 

9939 

994o 

994o 

5? 

3 

9748 

9749 

975o 

975i 

975i 

9752 

56 

3 

994o 

994i 

994i 

994i 

9942 

9942 

56 

4 

9753 

9753 

9754 

9755 

9756 

9756 

55 

4 

9942 

9943 

9943 

9943 

9944 

9944 

55 

5 

9757 

9758 

9758 

9759 

9760 

976o 

54 

5 

9944 

9945 

9945 

9945 

9946 

9946 

54 

6 

9761 

9762 

9763 

9763 

9764 

9765 

53 

6 

9946 

9947 

9947 

9947 

9948 

9948 

53 

7 

9765 

9766 

9767 

9767 

9768 

9769 

52 

7 

9948 

9949 

9949 

9949 

995o 

995o 

52 

8 

9769 

9770 

9771 

9772 

9772 

9773 

5i 

8 

995o 

9951 

995i 

r 

9951 

9952 

9952 

5i 

9 

9774 

9774 

9775 

9776 

9776 

9777 

5o 

9 

9952 

9953 

9953 

9953 

9953 

9954 

5o 

10 

9.999778 

9778 

9779 

978o 

9780 

978i 

49 

10 

9.999954 

9954 

9955 

9955 

9955 

9956 

49 

ii 

9782 

9782 

9783 

9784 

9784 

9785 

48 

ii 

9956 

9956 

9956 

9957 

9957 

9957 

48 

12 

9786 

9786 

9787 

9788 

9788 

9789 

47 

12 

9958 

9958 

9958 

9959 

9959 

9959 

47 

i3 

979° 

979° 

9791 

9792 

9792 

9793 

46 

i3 

9959 

9960 

996o 

996o 

996i 

996i 

46 

i4 

9794 

9794 

9795 

9795 

9796 

9797 

45 

i4 

9961 

9961 

9962 

9962 

9962 

9963 

45 

i5 

9797 

9798 

9799 

9799 

9800 

98oi 

44 

i5 

9963 

9963 

9963 

9964 

9964 

9o64 

44 

16 

9801 

9802 

98o3 

98o3 

98o4 

9804. 

43 

16 

9964 

9965 

9965 

9965 

9965 

9966 

43 

ll 

9805 

9806 

9806 

98o7 

9808 

9rfo8 

42 

T7 

9966 

9966 

9967 

9967 

9967 

9967 

42 

18 

9809 

9809 

9810 

98n 

9811 

9812 

4i 

18 

9968 

9968 

9968 

9968 

9969 

9969 

4i 

'9 

9813 

98i3 

98i4 

98i4 

98i5 

9816 

4o 

J9 

9969 

9969 

9970 

997° 

997° 

997° 

4o 

20 

9.999816 

9817 

9817 

98i8 

9819 

9819 

39 

20 

9-999971 

9971 

9971 

9971 

9972 

9972 

39 

21 

9820 

9820 

982I 

9822 

9822 

9823 

38 

21 

9972 

9972 

9973 

9973 

9973 

9973 

38 

22 

9824 

9824 

9825 

9825 

9826 

9827 

37 

22 

9973 

9974 

9974 

9974 

9974 

9975 

37 

23 

9827 

9828 

9828 

9829 

9829 

983o 

36 

23 

9975 

9975 

9975 

9976 

9976 

9976 

36 

24 

983i 

983i 

9832 

9832 

9833 

9834 

35 

24 

9976 

9976 

9977 

9977 

9977 

9977 

35 

25 

9834 

9835 

9835 

9836 

9836 

9837 

34 

25 

9977 

9978 

9978 

9978 

9978 

9979 

34 

26 

9838 

9838 

9839 

9839 

9840 

984o 

33 

26 

9979 

9979 

9979 

9979 

998° 

9980 

33 

27 

9841 

9842 

9842  9843 

9843 

9844 

32 

27 

998o 

9980 

998o 

998i 

998i 

998i 

32 

28 

9844 

9845 

9846  9846 

9847 

9847 

3i 

28 

998i 

9981 

9982 

9982 

9982 

9982 

0T 

<j  1 

29 

9848 

9848 

9849  9849 

985o 

985i 

3o 

29 

9982 

9983 

9983 

9983 

9983 

9983 

3o 

3o 

9.999851 

9852 

98529853 

9853 

9854 

29 

3o 

9.999983 

9984 

9984 

9984 

9984 

9984 

29 

3i 

9854 

9855 

98569856 

9857 

9857 

28 

3i 

9985 

9985 

9985 

9985 

9985 

9985 

28 

32 

9858 

9858 

9859;9859 

9860 

9860 

27 

32 

9986 

9986 

9986 

9986 

9986 

9986 

27 

33 

9861 

9861 

98629863 

9863 

9864 

26 

33 

9987 

9987 

9987 

9987 

9987 

9987 

26 

34 

9864 

9865 

9865*9866 

9866 

9867 

25 

34 

9988 

9988 

9988 

9988 

9988 

9988 

25 

35 

9867 

9868 

98689869 

9869 

9870 

24 

35 

9989 

9989 

V989 

9989 

9989 

9989 

24 

30 

9870 

9871 

9871  9872 

9872 

9873 

23 

36 

9989 

999° 

999° 

999° 

999° 

999° 

23 

37 

9873 

9874 

98749875 

9875 

9876 

22 

37 

999° 

999° 

999i 

9991 

999  l 

999  1 

22 

38 

9876 

9877 

9877^878 

9878 

9879 

21 

38 

9991 

999  1 

9991 

9992 

9992 

9992 

21 

39 

9879 

988o 

98809881 

9881 

9882 

2O 

39 

9992 

9992 

9992 

9992 

9992 

9998 

20 

4o 

9.999882 

9883 

9883  9884 

9884 

9885 

19 

4o 

9-999993 

9993 

9993 

9993 

9993 

9993 

'9 

4i 

9885 

9886 

9886  9887 

9887 

9888 

18 

4i 

9993 

9993 

9994 

9994 

9994 

9994 

18 

42 

9888 

9889 

98899890 

9890 

9891 

17 

42 

9994 

9994 

9994 

9994 

9994 

9995 

I7 

43 

9891 

9892 

9892!9892 

9893 

9893 

16 

43 

9995 

9995 

9995 

9995 

9995 

9995 

16 

44 
45 

9894 
9897 

9894 
9897 

9895  9895  9896 
9898|98989898 

9896 
9899 

i5 
i4 

44 
45 

9995 
9996 

9995 
9996 

9995 
9996 

9996 
9996 

9996 
9996 

9996 
9996 

i5 
i4 

46 

47 

9899 
9902 

99°° 
99o3 

9900  9901  9901 
99o3  99o3  9904 

9902 
9904 

i3 

12 

46 
4? 

9996 
9997 

9996 
9997 

9997 
9997 

9997 
9997 

9997 
9997 

9997 
9997 

i3 

12 

48 

99o5 

99o5 

9906  9906 

9906 

9907 

II 

48 

9997 

9997 

9997 

9998 

9998 

9998 

II 

49 

9907 

99o8 

9908  9909 

9909 

9910 

IO 

49 

9998 

9998 

9998 

9998 

9998 

9998 

10 

5o 

9.999910 

9910 

9911  9911 

99i2 

99I2 

9 

5o 

9-999998 

9998 

9998 

9998 

9998 

9998 

9 

5i 

99i3 

99i3 

99'3 

99i4 

99i4 

99i5 

8 

5i 

9999 

9999 

9999 

9999 

9999 

9999 

8 

52 

99  1  5 

99i5 

9916 

99i6 

9917 

99T7 

7 

52 

9999 

9999 

9999 

9999 

9999 

9999 

7 

53 

9918 

9918 

9918 

99i9 

9919 

9920 

6 

53 

9999 

9999 

9999 

9999 

9999 

9999 

6 

54 

992O 

9920 

9921 

992I 

9922 

9922 

5 

54 

9999 

9999 

9999 

9999 

9999 

5 

55 

9922 

9923 

9923 

9924 

9924 

9924 

4 

55 

o.oooooo 

oooo 

oooo 

oooo 

oooo 

oooo 

4 

56 

9925 

9925 

9926 

9926 

9926 

9927 

3 

56 

oooo 

oooo 

oooo 

oooo 

oooo 

oooo 

3 

5? 

9927 

9927 

9928 

9928 

9929 

9929 

2 

57 

oooo 

oooo 

oooo 

oooo 

oooo 

oooo 

2 

53 

9929 

993o 

993o 

993i 

993i 

993i 

I 

58 

oooo 

oooo 

oooo 

oooo 

oooo 

oooo 

I 

59 

9932 

9932 

9932 

9933 

9933 

9933 

O 

59 

oooo 

oooo 

oooo 

oooo 

oooo 

oooo 

O 

60" 

50" 

40" 

30" 

20" 

10" 

c 

60"     50" 

40" 

30"   20" 

10" 

ft 

Co-sine  of  1  Degree. 

>: 

Co-sine  of  0  Degree. 

1 

,  (  1"  2"  3"  4"  5"  6"  7"  8"  9" 
irt  $0000   00   001 

C  !//  2"  3"  4"  5"  6"  7"  8"  9" 

inl  oo  ooooooo 

LOGARITHMIC    TANGENTS. 


113 


F  

Tangent  of  88  Degrees. 

P.  Part 

3 

0" 

10" 

20" 

30" 

40" 

50" 

to  1". 

o 

11.456916 

ii  .457520 

ii.  458i25 

II.45873I 

11.459338 

II  .459945 

09 

60.6 

I 

46o553 

46n63 

461773 

462383 

462995 

4636o8 

58 

61.1 

a 

464221 

464836 

46545i 

466067 

466684 

4673o2 

57 

61.6 

3 

467920 

46854o 

469160 

469782 

470404 

47I027 

56 

62.2 

4 

47i65i 

472276 

472902 

473528 

474i56 

474785 

55 

62.7 

5 

4754i4, 

476o44 

476676 

477308 

47794i 

478575 

54 

63.3 

6 

479210 

479846 

480482 

481120 

481759 

482398 

53 

03.  8 

7 

483o39 

48368o 

484323 

484966 

4856n 

486256 

52 

64.4 

8 

486902 

487549 

488198 

488847 

489497 

490148 

5i 

65.o 

9 

490800 

49i453 

492107 

4.92762 

493418 

494o75 

5o 

65.5 

10 

ii.494?33 

11.495392 

it  .496052 

11.496713 

11.497375 

11.498038 

49 

66.1 

ii 

498702 

499367 

5ooo33 

500700 

5oi368 

5o2o37 

48 

66.8 

12 

502707 

5o3378 

5o4o5i 

504724 

5o5398 

506073 

47 

674 

i3 

506750 

507427 

5o8io6 

5o8785 

509466 

5ioi48 

46 

68.0 

i4 

5io83o 

5n5i4 

512199 

5i2885 

5i3572 

5i426o 

45 

68.6 

i5 

5i495o 

5i564o 

5i633i 

517024 

5i77i7 

5i84i2 

44 

69.3 

16 

519108 

SigSoS 

52o5o3 

52I2O2 

621903 

522604 

43 

70.0 

ll 

523307 

524OIO 

5247i5 

525421 

526128 

526837 

42 

70.7 

18 

527546 

528257 

528969 

629682 

53o396 

53ini 

4i 

7i.3 

'9 

53i828 

532545 

533264 

533984 

534705 

535428 

4o 

72.1 

20 

n.536i5i 

1  1.  536876 

.11  .537602 

ii.  53833o 

ii.539o58 

ii.539788 

3  9 

72.8 

21 

54o5i9 

54i25i 

541984 

542719 

543455 

544192 

38 

73.5 

22 

54493o 

54567o 

5464n 

547i53 

547896 

54864: 

37 

74.3 

23 

549387 

55oi34 

55o883 

55i632 

552384 

553i36 

36 

75.o 

24 

553890 

554645 

5554oi 

556i59 

5569i8 

557678 

35 

75.8 

25 

55844o 

5592o3 

559967 

560733 

56i5oo 

562268 

34 

•76.6 

26 

563o38 

563809 

56458: 

565355 

566i3o 

56690-7 

33 

777-5 

27 

567685 

568464 

569245 

570027 

570811 

57i596 

32 

78.3 

28 

572382 

573170 

573959 

574750 

575542 

576336 

3i 

79-- 

29 

577i3i 

5779a8 

578726 

579525 

58o326 

581128 

3o 

80.0 

3o 

11.581932 

11.582737 

u.583544 

n.584353 

n.585i63 

n.585974 

29 

80.9 

3i 

586787 

587601 

5884i7 

589235 

59oo54 

590874 

28 

81.8 

33 

591696 

592520 

593345 

594172 

595ooo 

59583o 

27 

82.8 

33 

596662 

597495 

598330 

599166 

600004 

6oo844 

26 

83.  7 

34 

6oi685 

602528 

603372 

604218 

6o5o66 

6o59i5 

25 

84-7 

35 

606766 

607619 

608474 

609330 

610187 

611047 

24 

85.7 

36 

611908 

612771 

6i3636 

6i45o2 

615370 

616240 

23 

86.7 

3? 

617111 

617985 

618860 

6i9737 

620615 

621496 

22 

87.8 

38 

622378 

623262 

624147 

625o35 

625924 

626815 

21 

88.8 

39 

627708 

628603 

629500 

630399 

63i299 

632201 

20 

89.9 

4o 

n.633io5 

ii  .634oi2 

ii  .634919 

n.635829 

11.636741 

n.637655 

I9 

91.1 

4i 

63857o 

639488 

640407 

641329 

642252 

643i77 

18 

92.2 

42 

644io5 

645o34 

645965 

646899 

647834 

648771 

17 

93.4 

43 

649711 

65o652 

65i595 

65254i 

653488 

654438 

16 

94.6 

44 

655390 

656343 

657299 

658257 

6592i7 

660179 

i5 

95.9 

45 

66  i  i  44 

662110 

663079 

664o5o 

665023 

665998 

i4 

97.2 

46 

666975 

667955 

668936 

669920 

670907 

671895 

i3 

98.5 

4? 

672886 

673879 

674874 

67587i 

676871 

677873 

12 

99.8 

48 

678878 

679885 

680894 

681905 

6829i9 

683935 

II 

101.3 

49 

684954 

685975 

686998 

688024 

689o52 

690083 

IO 

102.7 

5o 

11.691116 

n  .692151 

11.693189 

11.694230 

n.695273 

11.696318 

9 

IO4.2 

5i 

697366 

698417 

699470 

700526 

7oi584 

702645 

8 

105.7 

52 

703708 

704774 

705843 

706914 

707988 

709065 

7 

107.3 

53 

710144 

711226 

712311 

713398 

7i4488 

7i558i 

6 

108.9 

54 

716677 

717775 

718876 

719980 

721087 

•722196 

5 

no.  5 

55 

7233o9 

724424 

725542 

726663!   727787 

•728914 

4 

112  2 

56 

73oo44 

731176 

7323i2 

73345i!   734592 

735737 

3 

n4.o 

5? 

736885 

738o35 

739i89 

74o346 

74i5o6 

•742669 

2 

n5.8 

58 

743835 

745oo4 

746177 

747352 

74853i 

7497i3 

I 

"7-7  . 

59 

750898 

752087 

753279 

754474 

755672 

756874 

O 

119.7 

60'' 

50"         40" 

30" 

20"         10" 

. 

Co-  tangent  of  1  Degree.. 

* 

H 


114 


AUXILIARY    TABLE    FOR    SINES,    czc. 


1 

J 

0  Degree. 

1 

«        1  Degree. 

i 

3 

a 

s 

log.  sin.  A  — 
fog.  A1'. 

log.  tan.  A  — 
log.  A". 

log.  cot  A+ 
log.  A". 

1 

i 

9 

log.  sin.  A  —  log.  tan.  A  — 
log.  A".  1  log.  A". 

log.  cot  A  + 
log.  A". 

o 

0 

4.685575 

4.685575 

i5.3i4425 

60 

36oo 

0 

4.685553 

4.6856i9 

i5.3i438i 

60 

00 

I 

575 

575 

425 

59 

366o 

I 

552 

620 

38o 

59 

120 

2 

575 

575 

425 

58 

8720 

2 

55i 

622 

378 

58 

180 

3 

575 

575 

425 

57 

3780 

3 

55i 

623 

377 

57 

24o 

4 

575 

575 

425 

56 

384o 

4 

55o 

625 

375 

56 

3oo 

5 

575 

575 

425 

55 

39oo 

5 

549 

627 

373 

55 

36o 

6 

575 

575 

425 

54 

396o 

6 

548 

628 

372 

54 

420 

7 

575 

575 

4a5 

53 

4O2O 

7 

547 

63o    •   370 

53 

48o 

8 

574 

576 

424 

52 

4o8o 

8 

547 

632      368 

52 

54o 

9 

574 

576 

424 

5i 

4i4o 

9 

546 

6331     367 

5i 

600 

10 

4.685574 

4.685576 

i5.3i4424 

DO 

4200 

10 

4.685545 

4.685635 

i5.3i4365 

5o 

660 

ii 

5?4 

576 

424 

49 

4260 

1  1 

544 

637 

363 

49 

720 

12 

574 

577 

423 

48 

4320 

12 

543 

638 

362 

48 

780 

i3 

574 

577 

423 

47 

438o 

i3 

542 

64o 

36o 

47 

84o 

i4 

674 

577 

423 

46 

444o 

i4 

54i 

642 

358 

46 

900 

i5 

573 

578 

422 

45 

45oo 

i5 

54o 

644 

356 

45 

960 

16 

573 

578 

422 

44 

456o 

16 

539 

646 

354 

44 

IO2O 

J7 

573 

578 

422 

43 

4620 

17 

539 

647 

353 

43 

1080 

18 

573 

579 

421 

42 

468o 

18 

538 

649 

35i 

42 

u4o 

J9 

573 

579 

421 

4i 

474o 

J9 

537 

65i 

349 

4i 

I2OO 

20 

4.685572 

4.68558o 

i5.3i442o 

4o 

48oo 

20 

4.685536 

4.685653 

i5.3i4347 

4o 

1260 

21 

672 

58o 

420 

39 

486o 

21 

535 

655 

345 

39 

1320 

22 

672 

58i 

4i9 

38 

4920 

22 

534 

657 

343 

38 

i38o 

23 

672 

58i 

419 

37 

498o 

23 

533 

659 

34i 

37 

i44o 

24 

57i 

582 

4i8 

36 

5o4o 

24 

532 

661 

339 

36 

1600 

26 

671 

583 

4i7 

35 

5ioo 

25 

53i 

663 

337 

35 

1660 

26 

57i 

583 

4i7 

34 

5i6o 

26 

53o 

665 

335 

34 

1620 

27 

57o 

584 

4i6 

33 

522O 

27 

529 

668 

332 

33 

1680 

28 

57o 

584 

4i6 

32 

5280 

28 

527 

670 

33oj32 

1740 

-9 

57o 

585 

4i5 

3i 

534o 

29 

526 

672 

328 

3i 

1800 

3o 

4.685569 

4.685586 

i5.3i44i4 

3o 

54oo 

3o 

4.685525 

4.685674 

i5.3:4326 

3o 

1860 

3i 

569 

587 

4i3 

29 

546o 

3i 

524 

676 

324 

29 

1920 

32 

669 

587 

4i3 

28 

5520 

32 

523 

679 

321 

28 

1980 

33 

568 

588 

412 

27 

558o 

33 

522 

681 

3i9 

27 

2040 

34 

568 

589 

4n 

26 

564o 

34 

521 

683 

3i7 

26 

2IOO 

35 

567 

59o 

4io 

25 

5700 

35 

520 

685 

3i5 

25 

2l6o 

S6 

567 

59i 

409 

24 

5760 

36 

5i8 

688 

3l2 

24 

222O 

^7 

566 

592 

4o8 

23 

5820 

37 

5i7 

69o 

3io 

23 

228O 

38 

566 

593 

407 

22 

588o 

38 

5i6 

693 

3o7 

22 

2340 

39 

566 

593 

407 

21 

594o 

39 

5i5 

695 

3o5 

21 

24OO 

4o 

4.685565 

4.685594 

i5.3i44o6 

2O 

6000 

4o 

4.6855i4 

4.  685697 

i5.3i43o3 

20 

2460 

4i 

565 

595 

4o5 

I9 

6060 

4i 

5l2 

700 

3oo 

*9 

2626 

42 

564 

596 

4o4 

18 

6120 

42 

5n 

702 

298 

18 

2580 

43 

564 

598 

4O2 

*7 

6180 

43 

5io 

7o5 

295 

17 

2640 

44 

563 

599 

4oi 

16 

6240 

44 

509 

707 

293 

16 

2700 

45 

562 

600 

4oo 

i5 

63oo 

45 

5oy 

710 

290 

i5 

2760 

46 

562 

601 

399 

i4 

636o 

46 

5o6 

7i3 

287 

i4 

2820 

4? 

56i 

602 

398 

i3 

6420 

47 

5o5 

7i5 

285 

i3 

2880 

48 

56i 

6o3 

397 

12 

648o 

48 

5o3 

718 

282 

12 

2940 

49 

56o 

6o4 

396 

II 

654o 

49 

5o2 

720 

280 

II 

3ooo 

5o 

4.68556o 

4.6856o5 

i5.3i4395 

IO 

6600 

5o 

4.6855oi 

4.685723 

i5.3i4277 

10 

3o6o 

5i 

559 

607 

393 

9 

6660 

5i 

499 

726;     274 

9 

3l20 

62 

558 

608 

392 

6 

6720 

52l    498 

729      271 

8 

3i8o 

53 

558 

609 

39i 

7 

6780 

53 

497!     73i 

269 

7 

324o 

54 

557 

611 

389 

6 

684o 

54 

495 

734 

266 

6 

33oo 

55 

556 

612 

388 

5 

6900 

55 

494 

737 

263 

I 

336o 

56 

556 

6i3 

387 

4 

6960 

56 

492 

74o 

260 

4 

3420 

5? 

555 

6i5 

385 

3 

7020 

57 

491 

743 

257 

3 

348o 

58 

554 

616 

384 

2 

7080 

58 

49o 

745 

255 

2 

354o 

59 

554 

618 

382 

I 

7140 

59 

488 

748 

252 

I 

log.  cos.  A— 
log.  c.  A". 

log.  cot  A— 
log.  c.  A". 

log.  tan.  A+ 
log.  c.  A." 

1 

log.  cos.  A— 
log.  c.  A". 

log.  cot  A— 
log.  c.  A". 

log.  tail.  A-J- 
log.c.A" 

| 

89  Degrees. 

2 

88  Degrees. 

2 

LOGARITHMIC    TANGENTS. 


115 


1 

Tangent  of  89  Degrees. 

P.  Part 
tol". 

0" 

10" 

20" 

30" 

40" 

50" 

o 

11.758079 

11.759287 

II  .760498 

11.761714 

II  .762932 

H.764l54 

59 

121.7 

I 

765379 

766608 

767840 

769076 

77o3i5 

77i558 

58 

123.8 

2 

772805 

774o55 

775308 

776566 

777826 

779091 

57 

125.9 

3 

78o359 

78i63i 

782907 

784186 

785470 

786757 

56 

128.1 

4 

788047 

789342 

790641 

791943 

79325o 

79456o 

55 

i3o.4 

5 

79^874 

797192 

7985i5 

799841 

801171 

8o25o6 

54 

i32.8 

6 

8o3844 

805187 

8o6534 

8o7885 

809240 

810600 

53 

i35.3 

7 

811964 

8i3332 

814704 

816081 

817462 

818847 

52 

137.9 

8 

820237 

821632 

823o3i 

824434 

825842 

827255 

5i 

i4o.6 

9 

828672 

830094 

83i52o 

832951 

834387 

835828 

5o 

143.4 

10 

11.837273 

11.838724 

ii  .840179 

n.  841639 

n.843io4 

H.844574 

49 

146.2 

ii 

846o48 

847528 

849013 

85o5o3 

851999 

853499 

48 

149.3 

12 

855oo4 

8565i5 

858o3i 

859553 

861079 

862611 

47 

i52.4 

i3 

864149 

865692 

867240 

868794 

870354 

871919 

46 

i55.7 

i4 

873490 

875067 

876649 

878237 

879831 

88i43i 

45 

159.1 

i5 

883o37 

884648 

886266 

887890 

889519 

89n55 

44 

162.7 

16 

892797 

894446 

896100 

897761 

899429 

901103 

43 

166.4 

17 

902783 

904470 

906163 

.907863 

909569 

911283 

42 

170.3 

18 

9i3oo3 

914730 

916464 

918205 

919953 

921707 

4i 

174.4 

'9 

923469 

925239 

927015 

928799 

930590 

932388 

4o 

178.7 

20 

11.934194 

11.936008 

ii  .937829 

11.939658 

11.941494 

11.943338 

39 

i83.3 

21 

945191 

947o5i 

948919 

950795 

952679 

954572 

38 

188.0 

22 

956473 

958382 

960299 

962225 

964160 

966103 

37 

193.0 

23 

968o55 

970016 

971986 

973965 

975952 

977949 

36 

198.3 

24 

979956 

981971 

983996 

986o3o 

988074 

990128 

35 

2o3.9 

25 

992191 

994264 

996347 

998440 

I2.ooo543 

12.002657 

34 

209.8 

26 

12.004781 

12.006915 

12.009060 

12.01  I2l5 

oi3382 

oi5559 

33 

216.2 

27 

017747 

019946 

O22l56 

024378 

026611 

028855 

32 

222  7 

28 

oSini 

033379 

o35659 

O3795l 

o4o255 

042572 

3i 

229.8 

29 

044900 

047242 

049596 

051963 

o54342 

o56735 

3o 

237.3 

3o 

12.059142 

12  ,o6i56i 

12.063994 

I  2.  06644  I 

12.068902 

12.071377 

29 

245  4 

3i 

o73866 

076369 

078887 

O8l4l9 

083966 

086529 

28 

254.0 

32 

089106 

091699 

094308 

096932 

099572 

102228 

27 

263.2 

33 

104901 

107590 

110296 

iiSoig 

115760 

n85i7 

26 

273.2 

34 

121292 

I24o85 

126896 

129726 

i32574 

i  3544o 

25 

283.9 

35 

i38326 

i4i23i 

i44i56 

147100 

i5oo65 

i53o5o 

24 

295.4 

36 

i56o56 

159082 

i62i3o 

165199 

168290 

171404 

23 

3o8.o 

3? 

i7454o 

177698 

180880 

i84o85 

187314 

190567 

22 

321.7 

38 

i93845 

197148 

200476 

2o383o 

207210 

210616 

21 

336.7 

39 

214049 

217510 

220998 

2245i5 

228060 

23i635 

20 

353.1 

4o 

12.235239 

12.238873 

12.242538 

12.246235 

12.  249963 

12.253723 

19 

371.2 

4i 

257516 

261342 

265203 

269098 

273028 

276995 

18 

391.3 

4a 

280997 

285o37 

289115 

293232 

297388 

3oi584 

17 

4i3.6 

43 

3o582i 

Sioioo 

3l4422 

3i8787 

323i96 

327650 

16 

438.  7 

44 

332i5i 

336698 

341294 

345939 

35o634 

35538i 

i5 

467.0 

45 

36oi8o 

365o32 

369940 

374903 

379924 

385o64 

i4 

499.2 

46 

390143 

395345 

400609 

4o5938 

4n333 

416796 

i3 

536.2 

47 

422328 

427932 

4336io 

439362 

445192 

45noo 

12 

579.1 

48 

457091 

463i65 

469325 

475574 

481915 

488349 

II 

629.4 

49 

494880 

5oi5io 

5o8244 

5i5o83 

522O32 

529094 

IO 

689.4 

5o 

12.536273 

12.543572 

12.550996 

12.558549 

12.566236 

12.574061 

9 

5i 

582o3o 

590148 

598421 

6o6854 

6i5454 

624228 

8 

52 

633i83 

642327 

651667 

661212 

670972 

680956 

7 

53 

691175 

701641 

7i2365 

723360 

73464i 

746223 

6 

54 

758122 

77o357 

782946 

795911 

809275 

823o63 

5 

55 

8373o4 

802027 

867267 

883o6i 

899452 

9i6485 

4 

56 

9342i4 

952697 

972002 

992206 

i3.oi3395 

i3.  035671 

3 

57 

i3.o59i53 

1.3.083976 

i3.no3o5 

i3.i38334 

168297 

200482 

2 

58 

235244 

273032 

3i4425 

36oi83 

4n335 

469327 

I 

59 

536274 

6i5455 

712365 

8373o4 

i4-oi3395 

i4.3i4425 

O 

00" 

50" 

40" 

30" 

20" 

10" 

c 

Co-tangent  of  0  Degree. 

S 

116 


NATURAL    SINES. 


QE«! 

1° 

2° 

3° 

4° 

5° 

6° 

7° 

8° 

9° 

r-  — 

o 

000000 

017452 

034899 

o52336 

o69756 

087156 

104528 

I2i869 

i39i73 

i56434 

60 

I 

0291 

7743 

5i9o 

2626 

070047 

7446 

48i8 

2i58 

946i 

6722 

59 

2 

o582 

8o34 

548! 

2917 

o337 

7735 

5io7 

2447 

9749 

7oo9 

58 

3 

0878 

8325 

5772 

3207 

0627 

8o25 

5396 

2735 

i4oo37 

7296 

5? 

4 

n64 

8616 

6062 

3498 

0917 

83i5 

5686 

3024 

o325 

7584 

56 

5 

i454 

8907 

6353 

3788 

1207 

86o5 

5975 

33i3 

o6i3 

787i 

55 

6 

1745 

9T97 

6644 

4079 

1497 

8894 

6264 

36oi 

o9oi 

8i58 

54 

7 

2o36 

9488 

6934 

4369 

1788 

9i84 

6553 

389o 

n89 

8445 

53 

8 

2327 

9779 

7225 

466o 

2078 

9474 

6843 

4179 

i477 

8732 

52 

9 

2618 

020070 

75i6 

495o 

2368 

9763 

7l32 

4467 

i765 

9O2O 

5i 

10 

002909 

O2o36i 

037806 

o5524i 

072658 

o9oo53 

I0742I 

124756 

i42o53 

i593o7 

5o 

ii 

3200 

o652 

8097 

553i 

2948 

o343 

77io 

5o45 

a34i 

9594 

49 

12 

3491 

0942 

8388 

5822 

3238 

o633 

7999 

5333 

2629 

988i 

48 

i3 

3782 

1233 

8678 

6112 

3528 

O922 

8289 

5622 

29I7 

160168 

47 

i4 

4072 

i524 

8969 

6402 

38i8 

1212 

8578 

5910 

3205 

o455 

40 

i5 

4363 

i8i5 

9260 

6693 

4io8 

i5o2 

8867 

6199 

3493 

o743 

45 

16 

4654 

2106 

955o 

6983 

4399 

I79I 

9i56 

6488 

378o 

io3o 

44 

J7 

4945 

2397 

9841 

7274 

4689 

2081 

9445 

6776 

4o68 

i3i7 

43 

18 

5236 

2687 

o4oi32 

7564 

4979 

2371 

9734 

7o65 

4356 

1604 

42 

*9 

5527 

2978 

0422 

7854 

5269 

2660 

IIOO23 

7353 

4644 

i89i 

4i 

20 

oo58i8 

023269 

040713 

o58i45 

075559, 

o9295o 

Iio3i3 

127642 

i44932 

i62i78 

4o 

21 

6109 

356o 

1004 

8435 

5849 

3239 

0602 

793o 

5220 

.  2465 

39 

22 

6399 

385i 

1294 

8726 

6i39 

3529 

o89i 

8219 

55o7 

2752 

38 

23 

6690 

4i4i 

i585 

9&i6 

6429 

38i9 

1180 

85o7 

5795 

3o39 

37 

24 

6981 

4432 

1876 

93o6 

67:9 

4io8 

i469 

8796 

6o83 

3326 

36 

25 

7272 

4723 

2166 

9597 

7oo9 

4398 

i758 

9o84 

637i 

36i3 

35 

20 

7563 

5oi4 

2457   9887 

7299 

4687 

2047 

9373 

6659 

39oo 

34 

a7 

7»54 

53o5 

2748 

060177 

7589 

4977 

2336 

966i 

6946 

4i87 

33 

28 

8i45 

5595 

3o38 

o468 

7879 

5267 

2625 

9949 

7234 

4474 

32 

29 

8436 

5886 

3329 

0758 

8i69 

5556 

29l4 

i3o238 

7522 

476i|3i 

3o 

008727 

026177 

043619 

o6io49 

078459 

o95846 

Ii32o3 

i3o526 

i478o9 

i65o48 

3o 

3i 

9017 

6468 

3910 

i339 

8749 

6i35 

3492 

o8i5 

8o97 

5334 

20, 

3a 

93o8 

6759 

4201 

1620, 

9o39 

6425 

378i 

uo3 

8385 

56ai 

28 

33 

9599 

7049 

449i 

I92O 

9329 

67i4 

4070 

i39i 

8672 

59o8|27 

34 

9890 

7340 

4782 

22IO 

96l9 

7004 

4359 

1680 

896o 

6i95 

26 

35 

010181 

763i 

5072 

25oo 

99°9 

7293 

4648 

i968 

9248 

6482 

25 

36 

0472 

7922 

5363 

279I 

o8oi99 

7583 

4937 

2256 

9535 

6769 

24 

37 

0763 

8212 

5654 

3o8i 

o489 

7872 

5226 

2545 

9823 

7o56 

23 

38 

io54 

85o3 

5944 

33?I 

°779 

8162 

55i5 

2833 

iSoni 

7342 

22 

39 

1  344 

8794 

6235 

366i 

io69 

845i 

58o4 

3l2I 

o398 

7629 

21 

4o 

on635 

029085 

o46525 

063952 

o8i359 

o9874i 

n6o93 

i334io 

i5o686 

1679,16 

2O 

4i 

1926 

9375 

6816 

4242 

i649 

9o3o 

-  6382 

3698 

o973 

82o3 

19 

42 

2217 

9666 

7106 

4532 

i939 

9320 

6671 

3986 

1261 

8489 

18 

43 

25o8 

9957 

7397 

4823 

2228 

96o9 

696o 

4274 

1  548 

8776 

J7 

44 

2799 

o3o248 

7688 

5n3 

25i8 

9899 

7249 

4563 

i836 

9o63 

16 

45 

3090 

o539 

7978 

54c3 

2808 

100188 

7537 

485i 

2123 

935o 

i5 

46 

338o 

0829 

8269 

56g3 

3o98 

o477 

7826 

5i39 

2411 

9636 

i4 

47 

367i 

II2O 

8559 

5984 

3388 

0767 

8n5 

5427 

2698 

9923 

i3 

48 

3962 

i4i  i 

885o 

6274 

3678 

io56 

84o4 

57i6 

2986 

170209 

12 

49 

4253 

1702 

9140 

6564 

3968 

1  346 

8693 

6oo4 

3273 

0496 

II 

5o 

oi4544 

031992 

o4943i 

o66854 

o84258 

ioi635 

n8982 

l36292 

i5356i 

170783 

10 

5i 

4835 

2283 

9721 

-  ?i45 

4547 

I924 

9270 

658o 

3848 

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82° 

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80° 

d 

Natural  Co-sines. 

3 

££  4.85 

4.85 

4.84 

4.84 

4.83 

4.83 

4.82 

4.8i 

4.8o 

4.78 

IN  A  T  u  R  A  L    TANGENT  H 


11  V 


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88° 

87° 

86° 

85° 

84° 

83° 

82° 

81° 

80° 

£3 

Natural  Co-tangents. 

££'4.85  !  4.85 

4.86 

4.87 

4.88 

4.89 

4.9i 

4.93 

4.96 

4.98 

118 


NATURAL    SINES. 


.5 
| 

10° 

11° 

12° 

13° 

14° 

15° 

16° 

17° 

18° 

10° 

0 

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6668 

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9 

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10 

176512 

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244743 

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278432 

295l52 

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6798 

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8068 

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1908 

87i2|   543o 

2059 

8592 

49 

12 

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2335 

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7657 

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2751 

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6264 

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16 

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1  338 

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37 

24 

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25 

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3  1  20 

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8 

53 

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54 

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6 

55 

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4 

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1075 

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8187 

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3 

58 

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79° 

78° 

77° 

76° 

75° 

74° 

73° 

72C  !  71° 

70°  |  t 

Natural  Co-sines. 

">77 

4.75 

4.73 

4.7i 

4.69 

4.67 

4.65 

4.62 

4.6o 

4.57  | 

NATURAL    1"  A  N.G  E  N  T  s. 


119 


|   10° 

11° 

12° 

13° 

14° 

15° 

16° 

17° 

18° 

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37 

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5446 

43i6 

338i 

2606 

2i56 

36 

25 

3835 

1938 

220169 

854i 

7066 

5759 

4632 

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2979 

2483 

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26 

4i36 

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8848 

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3302 

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27 

4437 

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4737 

2847 

1084 

9464 

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558i 

4659 

3949 

3464 

32 

29 

5o38 

3i49 

1389 

9771 

8307 

7011 

5897 

4979 

4272 

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3i 

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185339 

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32 

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2610 

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6543 

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2916 

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7068 

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206483 

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261723 

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3i85oo 

337833 

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8157 

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3774 

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1087 

3oooi4 

9141 

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18 

43 

9253 

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5665 

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946i 

8806 

838o 

J7 

44 

9555 

7697 

597i 

439o 

2967 

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16 

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8000 

6277 

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II 

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igi363 

209518 

2278o6 

246241 

264834 

2836oo 

302553 

321707 

341077 

36o679 

10 

Si 

":865 

9822 

8112 

6549 

5i45 

39i4 

2870 

2028 

I4O2 

1008 

9 

62 

1966 

210126 

84i8 

6858 

5457 

4229 

3i88 

2349 

1727 

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8 

53 

2268 

0429 

8724 

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6768 

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7475 

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3823 

2991 

2377 

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6 

55 

2871 

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9337 

7784 

639i 

5172 

4i4i 

33i2 

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2324 

5 

56 

3i73 

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9643 

8092 

6702 

5487 

4459 

3634 

3027 

2653 

4 

57 

3475 

1  645 

9949 

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7014 

58oi 

4777 

SgSS 

3352 

2982 

3 

53 

3777 

1949 

230255 

87io 

7326 

6116 

5o95 

4277 

3677 

33i2 

2 

59 

4o78 

2253 

o562 

9019 

7637 

643i 

54i3 

4598 

4oi)2 

364i 

I 

79°  |  78°   77° 

76° 

75° 

74°   73° 

72°   71° 

70° 

a 

Natural  Co-tangents. 

§ 

to*r;.5-01  1  5'°5   5'C9 

5.i3 

5.i7 

5.22 

5.27 

5.33 

5.39   6.46 

120 


NATURAL    JS  i  N  E  a. 


A 
i 

20° 

21° 

22° 

23° 

24° 

25° 

20° 

27° 

28° 

29° 

o 

342020 

358368 

3746o7 

39o73i 

4o6737 

422618 

43837i 

453990 

469472 

4848  10 

60 

i 

2293 

864o 

4876 

0999 

•7002 

2882 

8633 

425o 

9728 

5o64 

59 

2 

2567 

8911 

5i46 

I267 

•7268 

3i45 

8894 

45o9 

9985 

53i8 

58 

3 

2840 

9i83 

54i6 

i534 

7534 

34o9 

9i55 

4768 

470242 

5573 

57 

4 

3n3 

9454 

5C85 

1802 

7799 

3673 

9417 

5o27 

o499 

5827 

56 

5 

3387 

9725 

5955 

2O70 

8o65 

3936 

9678 

5286 

o755 

6081 

55 

6 

366o 

9997 

6224 

2337 

833o 

4199 

9939 

5545 

IOI2 

6335 

54 

7 

3933 

360268 

6494 

26o5 

8596 

4463 

44O20O 

58o4 

1268 

659o 

53 

8 

4206 

o54o 

6763 

2872 

8861 

4726 

0462 

6o63 

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6844 

52 

9 

4479 

0811 

7o33 

3i4o 

9127 

4990 

0723 

6322 

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7o98 

5i 

10 

344752 

361082 

3773o2 

393407 

409392 

425253 

440984 

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472038 

487352 

5o 

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757i 

3675 

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6839 

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12 

5298 

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1896 

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4209 

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1767;   7357 

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5844 

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4477 

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2028 

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3320 

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16 

6390 

2709 

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Son 

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6832 

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8i33 

3576 

8875 

44 

17 

6663 

2980 

9187 

5278 

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2810 

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3832 

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18 

6936 

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22 

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2575 

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38 

23 

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0801 

6881 

2840 

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4375 

9942 

5368 

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37 

24 

8572 

4877 

1070 

7i48 

3io4 

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460200 

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36 

25 

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26 

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27 

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1  664 

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27 

34 

1297 

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3758 

9816 

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2780!   8181   3436 

26 

35 

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4027 

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6281 

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2658 

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4700 

21 

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5638 

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3397 

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42 

3475 

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1948 

7867 

3659 

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18 

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46 

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5872 

1244 

6469 

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4835 

1098 

7247 

3279 

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0618 

6129 

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48 

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3545 

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523i 

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6974 

12 

49 

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7784 

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n37 

6644 

2OO9 

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35565i 

3-71908 

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4o4o78 

419980 

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45i397 

466901 

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5i 

5923 

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8320 

4344 

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6017 

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7i58 

25i8 

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9 

52 

6194 

2448 

8588 

46  10 

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6278 

1916 

74i6 

2773 

7983 

8 

53 

6466 

27l8 

8856 

4876 

0772 

654o 

2I75 

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3028 

8236 

7 

54 

6738 

2988 

9124 

5i42 

io36 

6802 

2435 

793° 

3282 

8488 

6 

55 

7010 

3258 

9392 

54o8 

i3oo 

7o63 

2694 

8187 

3537 

874o 

5 

56 

7281 

3528 

9660 

5673 

i563 

7325 

2953 

8444 

3792 

8992 

A 

$7 

7553 

3797 

9928 

5939 

1827 

7587 

32i3 

8701 

4o46   9244 

3 

58 

7825 

4o67 

390196 

62o5 

2091 

7848 

3472 

8958 

43oi 

9496 

•i 

59 

80961   4337 

o463 

647i 

2355 

8110 

373i 

92l5 

4555 

9748 

i 

09°  |  68° 

67° 

66° 

65° 

64° 

63° 

62° 

61°   60° 

d 

Natural  Co-sines. 

££4.54  1  4.5i  i  4.48  1  4  45 

4.4i   4.38 

4.34 

4.3o 

4.26 

4.22  I 

NATURAL    TANGENTS, 


121 


1  A 
]  as 

20° 

21° 

22° 

23° 

24° 

25° 

26° 

27° 

28° 

29° 

o 

363970 

383864 

404026 

4s4475 

445229 

4663o8 

487733 

5o9525 

53i7o9 

5543o9 

60 

i 

43oo 

4198 

4365 

48:8 

5577 

6662 

8o93 

9892 

2o83 

4689 

59 

2 

4629 

4532 

47o3 

5i6s 

5926 

7016 

8453 

5io258 

2456 

6070 

58 

3 

49$9 

4866 

5o42 

55o5 

6275 

737i 

88i3 

0625 

2829 

545o 

57 

4   6288 

52OO 

538o 

5849 

6624 

7725 

9174 

O992 

32o3 

583i 

56 

5 

56  1  8 

5534 

57i9 

6192 

6973 

8080 

9534 

i359 

3577 

6212 

55 

6 

5948 

5868 

6o58 

6536 

7322 

8434 

9895 

I726 

395o 

6593 

54 

7 

6278 

6202 

6397 

6880 

767i 

8789 

49o256 

2093 

4324 

6974 

53 

8 

6608 

6536 

6736   -7224 

8020 

9i44 

0617 

2460 

4698 

7355 

52 

9 

6938 

6871 

7o75 

7568 

8369 

9499 

o978 

2828 

5o72 

7736 

5i 

10 

367268 

3872o5 

407414 

4279I2 

4487i9 

469854 

49i339 

5i3i95 

535446 

558ii8 

5o 

li 

7598 

754o 

7753 

8256 

9068 

47O2O9 

I7OO 

3563 

582i 

8499 

49 

12 

7928 

7874 

8092 

8601 

94i8 

o564 

2061 

393o 

6i95 

8881 

48 

i3 

8259 

8209 

8432 

8945 

9768 

O92O 

2422 

4298 

657o 

9*63 

47 

i4 

858g 

8544 

877i 

9289 

45on7 

1275 

2784 

4666 

6945 

9645 

46 

i5 

8919 

8879 

9111 

9634 

o467 

i63i 

3i45 

5o34 

73i9 

56ooa7 

45 

16 

925o 

9214 

945o 

9979 

o8i7 

i986 

35o7 

5402 

7694 

o4o9 

44 

17 

958i 

9% 

9790 

43o323 

n67 

2342 

3869 

577o 

8o69 

0791 

43 

18 

9911 

9884 

4ioi3o 

0668 

i5i7 

2698 

423l 

6i38 

8445 

n74 

42 

X9 

370242 

390219 

0470 

ioi3 

1868 

3o54 

4593 

65o7 

8820 

i556 

\i 

20 

370573 

39o554 

4io8io 

43i358 

452218 

4734io 

494955 

5i6875 

539i95 

561939 

4o 

21 

0904 

0889 

u5o 

1703 

2568 

0766 

53i7 

7244 

957i 

23?2 

39 

22 

1235 

1225 

1490 

2048 

29I9 

4l22 

5679 

76i3 

9946 

27o5 

38 

23 

i566 

i56o 

i83o 

2393 

3269 

4478 

6042 

,  7982 

54o322 

3o88 

37 

24 

1897 

1896 

2I7O 

2739 

3620 

4835 

64o4 

835i 

o698 

347i 

36 

25 

2228 

223l 

25ll 

3o84 

397i 

5i9i 

6767 

8720 

io74 

3854 

35 

26 

2559 

2567 

285i 

343o 

4322 

5548 

7i3o 

9089 

i45o 

4s38 

34 

27 

2890 

2903 

3192 

3775 

4673 

59o5 

7492 

9458 

1826 

4621 

33 

28 

3222 

3239 

3532 

4l2I 

5o24 

6262 

7855 

9828 

2203 

5oo5 

32 

29 

3553 

3574 

3873 

4467 

5375 

66i9 

8218 

52oi97 

2579 

5389 

3i 

3o 

3-/3885 

393910 

4i42i4 

434812 

455726 

476976 

498582 

520567 

542956 

565773 

3o 

3i 

4216 

4a47 

4554 

5i58 

6078 

7333 

8945 

o937 

3332 

6i57 

29 

32 

4548 

4583 

4895 

55o4 

6429 

769o 

93o8 

i3o7 

37o9 

654i 

28 

33 

488o 

4919 

5236|   585o 

6781 

8o47 

9672 

i677 

4o86 

6925 

27 

34 

6211 

5255 

5577 

6i97 

7l32 

84o5 

5ooo35 

2047 

4463 

73io 

26 

35 

5543 

5592 

59i9 

6543 

7484 

8762 

o399 

24l7 

484o 

7694 

25 

36 

5875 

5928 

6260 

6889 

7836 

9I20 

0763 

2787 

52i8 

80-79 

24 

3? 

6207 

6265 

6601 

7236 

8188 

9477 

1127 

3i58 

5595 

8464 

23 

38 

6539 

6601 

6943 

7682 

854o 

9835 

i49i 

3528 

5973 

8849 

22 

39 

6872 

6938 

-7284 

7929 

8892 

48oi93 

i855 

3899 

635o 

9234 

21 

4o 

3772o4 

397275 

41-7626 

438276 

459244 

48o55i 

5O22I9 

5242-70 

546728 

5696i9 

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4i 

7536 

7611 

7967 

8622 

9596 

o9o9 

2583 

464  1 

7106 

570004 

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42 

7869 

7948 

8309 

8969 

9949 

126-7 

2948 

6012 

7484 

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18 

43 

8201 

8285 

865i 

93i6 

46o3oi 

1626 

33i2 

5383 

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ll 

44 

8534 

8622 

8993 

9663 

o654 

1984 

3677 

5754 

8240 

1161 

16 

45 

8866 

8960 

9335 

44oou 

1006 

2343 

4o4i 

6i25 

86i9 

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46 

9199 

9297 

9677 

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27OI 

44o6 

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8997 

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47 

9532 

9634 

420019 

o7o5 

1712 

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477i 

6868 

9376 

23l9 

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48 

9864 

997i 

o36i 

io53 

2o65 

34i9 

5i36 

7240 

9755 

27o5 

12 

49 

380197 

4oo3o9 

070^ 

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2418 

3778 

55o2 

7612 

55oi34!   3o92 

II 

5o 

3?.o53o 

4oo64G 

421046 

441748 

462771 

484i37 

5o5867 

527984 

55o5i3 

673478 

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5i 

o863 

0984 

i389 

2095 

3124 

4496 

6232 

8356 

o892 

3865 

g 

52 

1196 

1322 

i73i 

2443 

3478 

4855 

6598 

8728 

I27I 

4252 

3 

53 

i53o 

1660 

2074 

279I 

383i 

52i4 

6963 

9ioo 

i65o 

4638 

7 

54 

i863 

I997 

24l7 

3i39 

4i85 

5574 

7329 

9473 

2030 

6026 

6 

55 

2196 

2335 

2759 

3487 

4538 

5933 

7695 

9845 

24o9 

54i3 

5 

56 

253o 

2673 

3102 

3835 

4892 

6293 

8061 

53o2i8 

2789 

58oo 

4 

67 

2863 

3on 

3445 

4i83 

5246 

6653 

8427 

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3i69 

6187 

3 

58 

3l97 

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3788 

4532 

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8793 

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3549 

6575 

2 

69 

353o 

3688 

4i32 

488o 

5954|   7373 

9i59 

i336 

3929 

6962 

I 

69° 

68° 

67° 

66° 

65° 

64° 

63° 

62° 

61° 

60° 

d 

Natural  Co-tangents. 

M 

s 

i 

.  1 

P.  P.  5  53 
to  I".5'" 

5.6o 

5.68 

5.76 

5.85 

5.95 

6.o5  [  6.16 

6.28 

6.4o 

1 

122 


NATURAL    SINES. 


n 

30'  |  31° 

32° 

33° 

34° 

35° 

36° 

37° 

38° 

39° 

o 

5ooooo 

5i5o38 

5299i9 

544639 

559i93 

573576 

587785 

6oi8i5 

6i566i 

629320 

60 

I 

O252 

5287 

53oi66 

4883 

9434 

38i5 

8021 

2o47 

589i 

9546 

59 

•2 

o5o4 

5537 

o4i2 

5l27 

9675 

4o53 

8256 

2280 

6120 

9772 

58 

3 

0706 

5786 

o659 

537i 

99i6 

4291 

849i 

25l2 

6349 

9998 

57 

4 

1007 

6o35 

0906 

56i5 

56oi57 

4529 

8726 

2744 

6578 

630224 

56 

5|   1269   6284 

rx5a 

5858 

o398 

4767 

896i 

2976 

6807 

o45o 

55 

6 

i5n 

6533 

i399 

6102 

o639 

5oo5 

9i96 

3208 

7036 

o676 

54 

7 

1762 

6782 

i645 

6346 

0880 

5243 

943i 

344o 

7265 

O9O2 

53 

8 

20l4 

7o3i 

1891 

6589 

II2I 

548r 

9666 

3672 

7494 

II27 

52 

9 

2266 

7280 

2i38 

6833 

i36i 

57i9 

99oi 

39o4 

7722 

i353 

5i 

10 

5o25i7 

517529 

532384 

547076 

56i6o2 

575957 

59oi36 

6o4i36 

6i795i 

63i578 

5o 

ii 

2769 

7778 

263o 

7320 

i843 

6i95 

o37i 

4367 

8180 

1804 

49 

12 

3020 

8027 

2876 

7563 

2o83 

6432 

0606 

4599 

84o8 

2O29 

48 

13 

3271 

8276 

3l22 

7807 

2324 

6670 

0840 

483i 

8637 

2255 

47 

i4 

3523 

8525 

3368 

8o5o 

2564 

69o8 

io75 

5o62 

8865 

2480 

46 

i5 

3774 

8773 

36i5 

8293 

28o5 

7i45 

i3io 

5294 

9o94 

27o5 

45 

16 

4o25 

9022 

386i 

8536 

3o45 

7383 

1  544 

5526 

9322 

293i 

44 

i? 

4276 

9271 

4io6 

8780 

3286 

7620 

i779 

5757 

955i 

3i56 

43 

18 

4528 

95i9 

4352 

9023 

3526 

7858 

20l3 

5988 

9779 

338i 

42 

'9 

4779 

9768 

4598 

9266 

3766 

8o95 

2248 

6220 

620007 

36o6 

4i 

20 

V>5o3o 

520016 

534844 

5495o9 

564007 

578332 

592482 

6o645i 

620235 

63383i 

4o 

21 

528i 

0265 

Sogo 

975a 

4247 

857o 

2716 

6682 

o464 

4o56 

39 

22 

5532 

o5i3 

5335 

9995 

4487 

8807 

295i 

6914 

o692 

4281 

38 

23 

5783 

0761 

558i 

55o238 

4727 

9o44 

3i85 

7i45 

O92O 

45o6 

37 

24 

6o34 

IOIO 

5827 

o48i 

4967 

928l 

34i9 

7376 

u48 

473i 

36 

25 

6285 

1258 

6072 

0724 

5207 

95i8 

3653 

76o7 

i376 

4955 

35 

26 

6535 

i5o6 

63  1  8 

o966 

5447 

9755 

3887 

7838 

1604 

5i8o 

34 

27 

6786 

1754 

6563 

I209 

5687 

9992 

4l2I 

8069 

i83i 

54o5 

33 

28 

7o37 

2OO2 

6809 

i452 

5927 

58o229 

4355 

83oo 

2o59 

5629 

32 

29 

7288 

225l 

7o54 

i694 

6166 

o466 

4589 

853i 

2287 

5854 

3r 

3o 

5o7538 

522499 

537300 

55i937 

5664o6 

580703 

594823 

6o876i 

622^15 

636078 

3o 

3i 

7789 

2747 

7545 

2180 

6646 

o94o 

5o57 

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2742 

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29 

32 

8o4o 

2995 

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2422 

6886 

1176 

529o 

9223 

2970 

6527 

28 

33 

8290 

3242 

8o35 

2664 

7125 

i4i3 

5524 

9454 

3i97 

675i 

27 

34 

854: 

3490 

8281 

2907 

7365 

i65o 

5758 

9684 

3425 

6976 

26 

35 

8791 

3738 

8526 

3i49 

7604 

1886 

599i 

99i5 

3652 

7200 

25 

36 

9041 

3986 

8771 

3392 

7844 

2123 

6225 

6ioi45 

388o 

7424 

24 

37 

9292 

4234 

9016 

3634 

8o83 

2359 

6458 

o376 

4io7 

7648 

23 

38 

9542 

448  1 

9261 

3876 

8323 

2596 

6692 

0606 

4334 

7872 

22 

39 

9792 

4729 

95o6 

4n8 

8562 

2832 

6925 

o836 

456i 

8096 

21 

4o 

5xoo43 

524977 

53975i 

55436o 

5688oi 

583o69 

597i59 

6no67 

624789 

638320 

20 

4i 

0293 

5224 

9996 

4602 

9o4o 

33o5 

7392 

i297 

5oi6 

8544 

19 

42 

o543 

5472 

540240 

4844 

928o 

354i 

7625 

l527 

5243 

8768 

18 

43 

o793 

57i9 

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5o86 

95l9 

3777 

7858 

i757 

547o 

8992 

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U 

io43 

5967 

0730 

5328 

9758 

4oi4 

8o92 

1987 

5697 

92l5 

16 

45 

1293 

6214 

o974 

5570 

9997 

425o 

8325 

2217 

5923 

9439 

i5 

46 

i543 

646  1 

I2I9 

58i2 

570236 

4486 

8558 

2447 

6i5o 

9663 

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47 

I793 

6709 

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6o54 

0475 

4722 

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2677 

6377 

9886 

i3 

48 

2o43 

6956 

1708 

6296 

0714 

4958 

0.024 

2907 

66o4 

64ono 

12 

49 

2293 

7203 

I953 

6537 

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5i94 

9256 

3i37 

683o 

o333 

II 

5o 

5i2543 

527450 

542197 

556779 

57ii9i 

585429 

599489 

6i3367 

627o57 

64o557 

10 

5i 

2792 

7697 

2442 

7021 

i43o 

5665 

9722 

3596 

7284 

0780 

9 

52 

3o42 

7944 

2686 

7262 

1669 

59oi 

9955 

3826 

75io 

ioc3 

8 

53 

3292 

8191 

2g3o 

75o4 

1907 

6i37 

600188 

4o56 

7737 

1226 

7 

54 

354i 

8438 

3i74 

7745 

2l46 

6372 

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4285 

7963 

i45o 

6 

55 

379i 

8685 

34i9 

7987 

2384 

6608 

o653 

45:5 

8189 

i673 

5 

5C 

4o4o 

8932 

3663 

8228 

2623 

6844 

o885 

4744 

84i6 

i896 

4 

5? 

4290 

9179 

39o7 

8469 

2861 

7°79 

1118 

4974 

8642 

2119 

3 

58 

4539 

9426 

4i5i 

8710 

3ioo 

73i4 

i35o 

52o3 

8868 

2342 

2 

59 

4789 

9673 

4395 

8952 

3338 

755o 

1  583 

5432 

9o94 

2565 

I 

59° 

58U   57° 

56° 

55° 

54°   53° 

52°   61° 

50° 

a 

Natural  Co-sines. 

S 

p  P 

!ol»>'8 

4.i3 

4.o9 

4-o4 

4.00 

3.95 

3.9o 

3.85   3.8o 

3.74 

1 

i\  A  T  u  u  A  L    TANGENT 


123 


!J 

30° 

31° 

32° 

33° 

34° 

35° 

36° 

37° 

38° 

39° 

1 

0 

5^735o 

600861 

624869 

649408 

674509 

700208 

726543 

753554 

781286 

8o9784 

Co 

I 

7738 

1257 

5274 

9821 

4932 

o64i 

6987 

4oio 

1754 

810266 

59 

2 

8126 

i653 

5679 

650235 

5355 

1075 

7432 

4467 

2223 

0748 

58 

3 

85i4 

2049 

6o83 

0649 

5779 

i5o9 

7877 

4923 

2692 

1230 

5? 

4 

8903 

2445 

6488 

io63 

6203 

i943 

8322 

538o 

3i6i 

1712 

56 

5 

9291 

2842 

6894 

i477 

6627 

2377 

8767 

5837 

363i 

2I95 

55 

6 

9680 

3239 

7299 

1892 

7o5i 

2812 

9213 

6294 

4ioo 

2678 

54 

7 

58oo68 

3635 

77°4 

23o6 

7475 

3246 

9658 

6751 

4570 

3i6i 

53 

8 

o457 

4o32 

8110 

2721 

7900 

368i 

73oio4 

7209 

5o4o 

3644 

52 

9 

o846 

4429 

85i6 

3i36 

8324 

4n6 

o55o 

7667 

55io 

4128 

5i 

JO 

58i235 

604827 

628921 

65355i 

678749 

7o455i 

73o996 

758i25 

785981 

814612 

5o 

rr 

i625 

5224 

9327 

3966 

9174 

4987 

J443 

8583 

645  1 

5o96 

49 

12 

2Ol4 

5622 

9734 

4382 

9599 

5422 

889 

9o4i 

6922 

558o 

48 

i3 

24o3 

6019 

63oi4o 

4797 

680025 

5858 

2336 

95oo 

7394 

6o65 

47 

i4 

2793 

6417 

o546 

52i3 

o45o 

6294 

2783 

9959 

7865 

6549 

46 

i5 

3i83 

68i5 

0953 

5629 

0876 

6730 

323o 

760418 

8336 

?o34 

45 

16 

35y3 

7213 

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6o45 

1302 

7166 

3678 

0877 

8808 

75l9 

44 

17 

3963 

7611 

1767 

646  1 

1728 

7603 

4i25 

i336 

928o 

8oo5 

43 

18 

4353 

8010 

2174 

6877 

2i54 

8o39 

4573 

i796 

975a 

849i 

42 

19 

4743 

84o8 

258i 

7294 

258o 

8476 

5O2I 

2256 

790225 

8976 

4i 

2P 

585i34 

608807 

632988 

657710 

683007 

7o89i3 

735469 

762716 

790697 

8i9463 

4o 

21 

5524 

9205 

3396 

8127 

3433 

935o 

59i7 

3i76 

1170 

9949 

39 

22 

SgiS 

9604 

38o4 

8544 

386o 

9788 

6366 

3636 

i643 

820435 

38 

23 

63o6 

6iooo3 

4ai  i 

896i 

4287 

710225 

68i5 

4o97 

2117 

O922 

37 

24 

6697 

o4o3 

4619 

9379 

47i4 

o663 

7264 

4558 

2590 

i4o9 

36 

25 

7088 

0802 

5027 

9796 

5i4a 

I  101 

77i3 

5oi9 

3o64 

i897 

35 

26 

7479 

I2OI 

5436 

660214 

5569 

i539 

8162 

548o 

3538 

2384 

34 

27 

7870 

1601 

5844 

o63i 

5997 

1977 

86n 

594i 

4012 

2872 

33 

28 

8262 

2001 

6253 

1049 

6425 

2416 

9061 

64o3 

4486 

336o 

32 

29 

8653 

24OI 

6661 

i467 

6853 

2854 

95n 

6865 

496i 

3848 

Si 

3o 

589045 

612801 

637070 

661886 

687281 

713293 

73996i 

767327 

795436 

824336 

3o 

3.1 

9437 

3201 

7479 

23o4 

7709 

3732 

74o4n 

7789 

59u 

4825 

29 

32 

9829 

36oi 

7888 

2723 

8i38 

4171 

0862 

8252 

6386 

53i4 

28 

33 

59022I 

4OO2 

8298 

3i4i 

8567 

46n 

l3l2 

87i4 

6862 

58o3 

27 

34 

o6i3 

4402 

8707 

356o 

8995 

5o5o 

I763 

9i77 

7337 

6292 

26 

35 

1006 

48o3 

9117 

3979 

9425 

5490 

22l4 

9640 

78i3 

6782 

25 

36 

i398 

52o4 

9527 

4398 

9854 

SgSo 

2666 

770104 

8290 

7272 

24 

37 

I79i 

56o5 

9937 

48i8 

690283 

637o 

3n7 

0567 

8766 

7762 

23 

38 

2184 

6006 

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5237 

0713 

6810 

3569 

io3i 

9242 

8252 

22 

39 

2577 

64o8 

o757 

5657 

n43 

725o 

4O2O 

i495 

97I9 

8743 

21 

4o 

59297o 

616809 

641167 

666077 

691572 

717691 

744472 

77i959 

800196 

829234 

2O 

4i 

3363 

7211 

i578 

6497 

2003 

8:32 

4925 

2423 

0674 

9725 

19 

42 

3757 

76i3 

1989 

6917 

2433 

8573 

5377 

2888 

u5i 

83o2i6 

18 

43 

4i5o 

8oi5 

2399 

7337 

2863 

9014 

583o 

3353 

i629 

0707 

I7 

44 

4544 

8417 

2810 

7758 

3294 

9455 

6282 

38i8 

2107 

1199 

16 

45 

4937 

8819 

3222 

8179 

3725 

9897 

6735 

4283 

2585 

1691 

i5 

£6 

533i 

9221 

3633 

8599 

4i56 

720339 

7i89 

4748 

3o63 

2i83 

i4 

47 

5725 

9624 

4o44 

9020 

4587 

0781 

7642 

52i4 

3542 

2676 

i3 

48 

6  120 

620026 

4456 

9442 

5oi8 

1223 

8o96 

568o 

4O2I 

3169 

12 

49 

65i4 

0429 

4868 

9863 

545o 

i665 

8549 

6  1  46 

45oo 

3662 

II 

5o 

596908 

620832  645280 

670284 

695881 

722108 

749003 

776612 

8o4979 

834x55 

IO 

5i 

73o3 

12351   5692 

0706 

63i3 

255o 

9458|   7078 

5458 

4648 

o 

52 

7698 

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1128 

6745 

2993 

9912   7545 

5938 

5x42 

g 

53 

8093 

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i55o 

7177 

3436 

75o366|   8012 

64i8 

5636 

7 

54 

8488 

2445 

6929 

1972 

7610 

3879 

082  i  1   8479 

6898 

6i3o 

6 

55 

8883 

2849 

7342 

2394 

8042 

4323 

1276 

8946 

7379 

6624 

5 

56 

9278 

3253 

7755 

2817 

8475 

4766 

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94i4 

7859 

7119 

4 

57 

^  9674 

3657 

8168 

324o 

8908 

52IO 

2187 

9881 

834o 

7614 

3 

58 

600069 

4o6i 

858i 

3662 

934i 

5654 

2642 

780349 

8821 

8109 

2 

59 

o465 

4465 

8994 

4o85 

9774 

6o98 

3098 

0817 

93o3 

86o4 

I 

59° 

58° 

57° 

56° 

55° 

54° 

53° 

52° 

51° 

50° 

J 

Natural  Co-tangents. 

1 

£#6.53  |  6.67 

6.82 

6.97 

7-i4 

7.3i 

7.5o 

7.70  7.92   s.i4 

124 


NAIURAI.    SINES. 


d 

i 

40° 

41° 

42° 

43° 

44° 

45° 

46° 

47° 

48° 

49° 

) 

0 

642788 

656o59 

669i3i 

68i998 

C94658 

7o7io7 

7i934o 

73i354 

743i45 

7547io 

60 

I 

3oio 

6279 

9347 

2211 

4868 

73l2 

9542 

i552 

3339 

49oo 

59 

2 

3233 

6498 

9563 

2424 

5o77 

75i8 

9744 

1700 

3534 

5o9i 

58 

3 

3456 

6717 

9779 

2636 

5286 

7724 

9946 

i949 

3728 

5282 

57 

6 

3679 

6937 

9995 

2849 

5495 

•7929 

720148 

2147 

3923 

5472 

56 

5 

3901 

7i56 

67O2II 

3o6i 

57o4 

8i34 

o349 

2345 

4117 

5663  55 

6 

4124 

7375 

0427 

3274 

59i3 

834o 

o55i 

2543 

43i2 

5853  54 

7 

4346 

7594 

0642 

3486 

6122 

8545 

o753 

2741 

45o6 

6o44!53 

8 

4569 

7814 

o858 

3698 

633o 

875o 

o954 

2939 

4700 

6234 

52 

9 

4791 

8o33 

1074 

39n 

6539 

8956 

n56 

3i37 

4894 

6425 

5i 

10 

645oi3 

658252 

67i289 

684i23 

696748 

7o9i6i 

721357 

733334 

745o88 

7566i5 

5o 

ii 

5236 

847i 

i5o5 

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6957 

9366 

i559 

3532 

5282 

68o5 

49 

12 

5458 

8689 

1721 

4547 

7i65 

957i 

1760 

373o 

5476 

6995 

48 

i3 

568o 

8908 

i936 

4759 

7374 

9776 

I962 

3927 

567o 

7i85 

47 

i4 

59O2 

9I27 

2l5l 

497i 

7582 

998i 

2i63 

4i25 

5864 

7375 

46 

i5 

6124 

9346 

2367 

5i83 

779° 

710185 

2364 

4323 

6o57 

7565 

45 

16 

6346 

9565 

2582 

5395 

7999 

o39o 

2565 

4520 

625i 

7755 

44 

17 

6568 

9783 

2797 

56o7 

8207 

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2766 

4717 

6445 

7945 

43 

18 

6790 

660002 

3oi3 

58i8 

84i5 

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2967 

49i5 

6638 

8i34 

42 

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7012 

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3228 

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8623 

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3i68 

5lI2 

6832 

8324!4i 

20 

647233 

66o439 

673443 

686242 

C98832 

711209 

723369 

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747025 

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4o 

21 

7455 

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3658 

6453 

9o4o 

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357o 

55o6 

7218 

87o3 

39 

22 

7677 

0875 

3873 

6665 

9248 

1617 

377i 

57o3 

7412 

8893 

38 

23 

7898 

109.4 

4o88 

6876 

9455 

1822 

39?I 

59oo 

7605 

9082 

37 

24 

8120 

l3l2 

4302 

7088 

9663 

2026 

4172 

6o97 

7798 

927I 

36 

25 

834i 

i53o 

45i7 

7299 

987i 

2230 

4372 

6294 

799  1 

946i 

35 

26 

8563 

i?48 

4732 

75io 

7ooo79 

2434 

4573 

649i 

8184 

965o 

34 

27 

8784 

1966 

4947 

7721 

0287 

2639 

4773 

6687 

8377 

9839 

33 

28 

9006 

2184 

5i6i 

7932 

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2843 

4974 

6884 

857o 

760028 

32 

29 

9227 

2402 

5376 

8i44 

0702 

3o47 

5i74 

7081 

8763 

0217 

3i 

3o!  649448 

662620 

67559o 

688355 

7oo9o9 

7i325o 

725374 

737277 

748956 

760406 

3o 

3i 

9669 

2838 

58o5 

8566 

1117 

3454 

5575 

7474 

9i48 

o595 

2O 

32 

9890 

3o56 

6oi9 

8776 

1  324 

3658 

5775 

7670 

934i 

0784 

2o 

33 

65om 

3273 

6233 

8087 

i53i 

3862 

5975 

7867 

9534 

?972 

27 

34 

o332 

349i 

6448 

9i98 

I739 

4o66 

6175 

8o63 

9726 

1161 

26 

35 

o553 

37o9 

6662 

94o9 

i946 

4269 

6375 

8259 

99i9 

i35o 

25 

36 

0774 

3926 

6876 

9620 

2i53 

4473 

6575 

8455 

75oi  i  i 

i538 

24 

3? 

o995 

4i44 

7O9o 

983o 

236o 

4676 

6775 

865i 

o3o3 

1727 

23 

38 

1216 

436i 

73o4 

69oo4i 

2567 

488o 

6974 

8848 

o496 

I9i5 

22 

39 

i437 

4579 

75i8 

O25l 

2774 

5o83 

7174 

9o43 

0688 

2104 

21 

4o 

65i657 

664796 

677732 

69o462 

702981 

7i5286 

727374 

739239 

750880 

762292 

20 

4i 

1878 

5oi3 

7946 

0672 

3i88 

5490 

7573 

9435 

1072 

2480 

*9 

42 

2098 

523o 

8160 

0882 

SSgS 

5693 

7773 

963i 

1264 

2668 

18 

43 

2319 

5448 

8373 

io93 

36oi 

5896 

7972 

9827 

i456 

2856 

i? 

44 

2539 

5665 

8587 

i3o3 

38o8 

6099 

8172 

740023 

1  648 

3o44 

16 

45 

2760 

5882 

8801 

i5i3 

4oi5 

6302 

8371 

0218 

1840 

3232 

i5 

46 

2980 

6099 

9oi4 

1723 

4221 

65o5 

857o 

o4i4 

2032 

3420 

i4 

47 

32OO 

63i6 

9228 

i933 

4428 

67o8 

8769 

o6o9 

2223 

36o8 

i3 

48 

342i 

6532 

944i 

2i43 

4634 

691  1 

8969 

o8o5 

24i5 

3796 

12 

49 

364i 

6749 

9655 

2353 

484  1 

7n3 

9i68 

IOOO 

2606 

3984 

II 

5o 

65386i 

666066 

679868 

692563 

7o5o47 

7i73i6 

729367 

74n95 

752798 

764171 

10 

5i 

4o8i 

7i83 

680081 

2773 

5253 

75l9 

9566 

i39i 

2989 

4359 

9 

52 

43oi 

7399 

0295 

2983 

5459 

7721 

9765 

1  586 

3i8i 

4547 

8 

53 

452i 

7616 

o5oS 

3l92 

5665 

7924   9963 

1781 

3372 

4734 

7 

54 

4?4i 

7833 

0721 

3402 

5872 

8126  73oi62 

I976 

3563 

4921 

6 

55 

4961 

8o49 

o934 

36n 

6078 

8329>   o36i 

2171 

3755 

5  1  09 

5 

56 

5i8o 

8265 

ii47 

382i 

6284 

853i 

o56o 

2366 

3946 

5296 

4 

57 

54oo 

8482 

i36o 

4o3o 

6489 

8733 

0758 

256i 

4i37 

5483 

3 

58 

5620 

8698 

i573 

424o 

6695 

8936 

c957 

2755 

4328 

5670 

2 

59 

5839 

89i4 

i786 

4449 

6901 

9i38 

n55 

295o 

45i9 

5857 

I 

49° 

48° 

47° 

46° 

45° 

44° 

43° 

42° 

41° 

40° 

5 

Natural  Co-sines. 

3* 

££3.69)  3.63  |  3.57 

3.5a 

3.46 

3.4o 

3.34 

3.27 

3.21 

3.i5 

i 

J 

NATURAL    TANGENTS 


125 


4 

i 

40° 

41°  '  42° 

43° 

44° 

45° 

46° 

47° 

48° 

49° 

1 

0 

839ioo 

869287  9oo4o4, 

9325i5 

965689 

I.OOOOO 

i.o3553 

1.07237 

i.  11061 

I.i5o37 

60 

I 

9595 

9798 

o93i)   3o59 

625i 

oo58 

36i3 

7299 

1126 

5io4 

59 

2 

840092 

870309 

i458 

36o3 

68i4 

0116 

3674 

7362 

1191 

5172 

58 

3 

o588 

0820 

i985 

4i48 

7377 

oi75 

3734'   7425 

1256 

524o 

57 

4 

1084 

i332 

25i3 

4693 

794o 

0233 

3794 

7487 

l32I 

53o8 

56 

5 

i58i 

i843 

3o4i 

5238 

85o4 

O29I 

3855 

755o 

i387 

5375 

55 

6 

2078 

2356 

3569 

5783 

9o67 

o35o 

39i5 

7613 

i452 

5443 

54 

7 

2575 

2868 

4o98 

6329 

9632 

o4o8 

3976 

7676 

i5i7 

55n 

53 

8 

3o73 

338i 

4627 

6875 

97oi96 

o467 

4o36 

7738 

l5&2 

5579 

52 

9 

357i 

3894 

5i56 

7422 

0761 

o525 

4097 

7801 

1  648 

5647 

5i 

10 

844069 

874407 

9o5685 

937968 

97i326 

i.oo583 

I.o4i58 

1.07864 

1.11713 

i  .:57i5 

5o 

ii 

4567 

4920 

62i5 

85i5 

1892 

0642 

4218 

7927 

i778 

5783 

49 

12 

5o66 

5434 

6745 

9063 

2458 

0701 

4279 

799° 

1  844 

585i 

48 

i3 

5564 

5948 

7275 

9610 

3o24 

0759 

434o 

8o53 

J9°9 

59i9 

47 

i4 

6o63 

6462 

78o5 

94oi58 

3590 

0818 

44oi 

8116 

i975 

5987 

46 

i5 

6562 

6976 

8336 

0706 

4i57 

0876 

446  1 

8179 

2041 

6o56 

45 

16 

7062 

7491 

8867 

1255 

4724 

o935 

4522 

8243 

2106 

6124 

44 

J7 

7562 

8006 

9398 

i8o3 

5291 

o994 

4583 

83o6 

2I72 

6192 

43 

18 

8062 

852i 

993o 

2352 

5859 

io53 

4644 

8369 

2238 

6261 

42 

i9 

8562 

9°37 

2902 

6427 

III2 

47o5 

8432 

23o3 

6329 

4x 

20 

849062 

879553 

9io994 

94345i 

976996 

I.OII70 

1.04766 

1.08496 

i.i2369 

i.i6398 

4o 

21 

9563 

880069 

i526 

4ooi 

7564 

I229 

4827 

8559 

2435 

6466 

39 

22 

85oo64 

o585 

2o59 

4552 

8i33 

1288 

4888 

8622 

25oi 

6535 

38 

23 

o565 

1102 

2592 

5  1  02 

87o3 

1  347 

4949 

8686 

2567 

66o3 

37 

24 

1067 

1619 

3i25 

5653 

9272 

i4o6 

5oio 

8749 

2633 

6672 

36 

25 

i568 

2i36 

3659 

8204 

i465 

5072 

88i3 

2699 

674i 

35 

26 

2070 

2653 

4193 

6756 

98o4i3 

i524 

5i33 

8876 

2765 

6809 

34 

27 

2573 

3i7i 

4727 

73o7 

o983 

i583 

5194 

894o 

283i 

6878 

33 

28 

3o75 

3689 

5261 

7859 

i554 

1642 

5255 

9oo3 

2897 

6947 

32 

29 

3578 

4207 

5796 

8412 

2126 

1702 

53i7 

9o67 

2963 

7016 

3i 

3o 

854o8i 

884725 

9i633i 

948965 

982697 

1.01761 

1.05378 

I.o9i3i 

1.13029 

1.17085 

3o 

3i 

4584 

5244 

6866 

95i8 

3269 

1820 

5439 

9i95 

3096 

7i54 

29 

32 

5o87 

5763 

7402 

95oo7i 

3842 

1879 

55oi 

9258 

3i62 

7223 

28 

33 

559i 

6282 

7938 

0624 

44i4 

I939 

5562 

9322 

3228 

7292 

27 

34 

6o95 

6802 

8474 

n78 

4987 

1998 

5624 

9386 

3295 

736i 

26 

35   6599 

7321 

9oio 

i733 

556o 

2057 

5685 

945o 

336i 

743o 

25 

36   7io4 

7842 

9547 

2287 

6i34 

2117 

5?47 

95i4 

3428 

75oo 

24 

37j  -7608 

8362 

920084 

2842 

6708 

2176 

58o9 

9578 

3494 

7569 

23 

38 

8n3 

8882 

0621 

3397 

7282 

2236 

587o 

9642 

356i 

7638 

22 

39 

86i9 

94o3 

1159 

7857 

2295 

5932 

97°6 

3627 

7708 

21 

4o 

859124 

889924 

921697 

9545o8 

988432 

1.02355 

i.o5994 

i.i3694 

i.i7777 

2O 

4i 

963o 

890446 

2235 

5o64 

9oo7 

24i4 

6o56 

'  9834 

376i 

7846 

'9 

42 

86oi36 

0967,   2773 

562i 

9582 

2474 

6117 

9899 

3828 

7916 

18 

43 

0642 

1489!   33i2 

6177 

99oi58 

2533 

6179 

o963 

3894 

7986 

17 

44 

n48 

2OI2 

385i 

6734 

o735 

2593 

6241 

1.10027 

396i 

8o55 

16 

45 

i655 

2534 

439o 

7292 

i3n 

2653 

63o3 

oo9i 

4028 

8i25 

i5 

46 

2162 

3o57 

493o 

7849 

1888 

2713 

6365 

oi56 

4o95 

8194 

i4 

47 

2669 

358o 

5470 

8407 

2465 

2772 

6427 

O22O 

4162 

8264 

i3 

48 

3i77 

4io3J   6oioj   8966 

3o43 

2832 

6489 

0285 

4229 

8334 

12 

49 

3685 

4627 

655i 

9524 

362i 

2892 

655i 

o349 

4296 

84o4 

II 

5o 

864i93 

895i5i 

927091 

96oo83 

994i99 

I  O2952 

i.o66i3 

I.io4i4 

1.  14363 

i.i8474 

10 

5i 

47oi 

5675j   7632 

0642 

4778 

3012 

6676 

0478 

443o 

8544 

9 

52 

5209 

6199   8174 

I2O2 

5357 

3072 

6738 

o543 

4498 

86i4 

8 

53 

57i8 

6724!   8715 

1761 

5936 

3i32 

6800 

0607 

4565 

8684 

7 

54 

6227 

7249   9257 

2322 

65:5 

3l92 

6862 

0672 

4632 

8754 

6 

55 

6736 

7774 

9800 

2882 

7°95 

3252 

6925 

o737 

4699 

8824 

5 

56 

7246 

8299 

93o342 

3443 

7676 

33i2 

6087 

0802 

4767 

8894 

4 

57 

7756 

8825 

o885 

4oo^ 

8256 

3372 

7o49 

o867 

4834 

8964 

3 

58 

8266 

935i 

1428 

4565 

8837 

3433 

7112 

o93i 

4902 

9o35 

2  ! 

59 

8776 

9877 

1971 

5l27 

94i8 

3493 

7i74 

0996 

4969 

9io5  i  1 

49° 

48° 

47° 

•  46° 

45° 

44° 

43° 

42°    41° 

-5   1 

Natural  Co-tangents. 

PT> 
•ft  o  o 
to  1" 

8.64 

8.92 

0.21 

9.53 

°-99 

i  .02 

i  .06 

I  .  IO 

i.i5 

1 

L  :. 

126 


A  T  U  R  A  L, 


.4 

a 

50C 

51° 

52° 

53° 

54° 

55° 

56° 

57° 

5S°  ' 

59° 

o 

766044 

777146 

•788011 

798636 

8o9oi7 

8i9i52 

829o38 

838671 

848o48 

857i67 

60 

i 

623i 

7329 

8i9o 

8811 

9i88 

93i9 

9200 

«829 

8202 

73i7 

59 

2 

64i8 

75i2 

8369 

8985 

9359 

9486 

9363 

8987 

8356 

7467 

58 

3 

66o5 

7695 

8548 

9i6o 

953o 

9652 

9525 

9146 

85io 

7616 

5? 

4 

6792 

7878 

8727 

9335 

97oo 

98i9 

9688 

93o4 

8664 

7766 

56 

5 

6979;   8060 

89o5 

95io 

987i 

9985 

985o 

9462 

8818 

79i5 

55 

6 

7i65 

8243 

9o84 

9685 

810042 

820152 

83ooi2 

0.620 

8972 

8o65 

54 

7 

7352 

8426   9263 

9859 

O2I2 

o3i8 

oi74 

9778 

9I25 

8214 

53 

8 

7538 

86o8|   944i 

8ooo34 

o383 

o485 

o337 

9936 

9279 

8364|5a 

9 

7725 

8791 

9620 

0208 

o553 

o65i'  o499 

84oo94 

9433 

85i3l5i 

10 

767911 

778973 

789798 

8oo383 

8io723 

820817*830661 

84oa5i 

849586 

858662 

5o 

ii 

8097 

9i56 

9977 

o557 

o894 

o983 

0823 

o4o9 

9739 

88zi 

49 

12 

8284 

9338 

79oi55 

o73i 

1064 

n49 

o984 

o567 

9893 

896o 

48 

i3 

8470 

952O 

o333 

o9o6 

1234 

i3i5 

n46 

0-724 

85oo46 

9io9 

47 

i4 

8656 

9702 

o5n 

1080 

i4o4 

i48i 

i3o8 

0882 

oi99 

9258 

46 

16 

8842 

9884 

o69o 

1254 

i574 

i647 

i47o 

•  io39 

o352 

94o6 

45 

16 

9028 

780067 

0868 

1428 

i744!  i8i3 

i63i 

n96 

o5o5 

9555 

44 

I7 

9214 

0249 

1046 

1602 

i9i4 

I978 

i793 

i354 

o658 

97o4 

43 

18 

9400 

o43o 

1224 

1776 

2084 

2i44 

i954 

i5n 

0811 

9S52 

42 

J9 

9585 

0612 

i4oi 

i949 

2253 

23lO 

2Il5 

i668j   o964 

860001 

4i 

20 

769771 

78o794 

79l579 

802123 

812423 

822475 

832277 

84i825 

85ni7 

86oi49 

4o 

21 

9957 

o976 

i757 

2297 

2592 

264i;  2438 

I982 

I269 

0297 

39 

22 

770142 

1167 

i935 

2470 

2762 

2806 

2599 

2i39 

1422 

o446 

38 

23 

o328 

i339 

21  12 

2644 

293l 

297I 

276o 

2296 

i575 

o594 

37 

24 

o5i3 

l520 

2290 

2817 

3ioi 

3i36 

292I 

2452 

1727 

0742 

36 

25 

0699 

1702 

2467 

299I 

3270 

33o2 

3o82 

26o9 

1879 

o89o 

35 

26 

o884 

i883 

2644 

3i64 

3439 

3467 

3243 

2-766 

2032 

io38 

34 

27 

1069 

2o65 

2822 

3337 

36o8 

3632 

34o4 

2922 

2184 

1186 

33 

28 

I254!    2246 

2999 

35n 

3778 

3797 

3565 

3o79 

2336 

i334 

32 

29 

i44o 

2427 

3i76 

3684 

3947 

396i 

3725 

3235 

2488 

1481, 

3i 

3o 

771625 

782608 

793353 

8o3857 

8i4n6 

824126 

833886 

84339i 

85264o 

861620' 

3o 

3i 

1810 

2789 

353o 

4o3o 

4284 

429I 

4o46 

3548 

2792 

1777 

29 

32 

i995 

2970 

37o7 

4203 

4453 

4456 

4207 

37o4 

2944 

I924 

28 

33 

2179 

3i5i 

3884 

4376 

4622 

4620 

4367 

386o 

3o96 

2072 

27 

34 

2364 

33321   4o6i 

4548 

479i 

4785 

4527 

4oi6 

3248 

22I9 

26 

35 

2549 

35i3 

4238 

4721 

4959 

4949 

4688 

4172 

3399 

2366 

25 

36 

2734 

3693 

44i5 

4894 

5128 

5n3 

4848 

4328 

355i 

25i4 

24 

3? 

2918 

3874 

459i 

5o66 

5296 

5278 

5oo8 

4484 

3702 

2661 

23 

38 

3I03 

4o55 

4768 

5239 

5465 

5442 

5i68 

464o 

3854 

2808 

22 

39 

3287 

4235 

4944 

54n 

5633 

56o6 

5328 

4795 

4oo5 

2955 

21 

4o 

773472 

784416 

795i2i 

8o5584 

8i58oi 

82577o 

835488 

84495i 

854i56 

863io2 

2O 

4i 

3656 

4596 

5297 

5756 

5969 

5934 

5648 

5io6 

43o8 

3249 

J9 

42 

384o 

4776 

5473 

5928 

6i38 

6o98 

58o7 

5262 

4459 

3396 

18 

43 

4024 

4957 

565o 

6100 

63o6 

6262 

5967 

54i7 

4610 

3542 

i? 

44 

4209 

5i37 

5826 

6273 

6474 

6426 

6127 

5573 

4-761 

3689 

16 

45 

4393 

53i7 

6002 

6445 

6642 

659o 

6286 

5728 

49I2 

3836 

i5 

46 

4577 

5497 

6178 

6617 

68o9   6753 

6446 

5883 

5o63 

3982 

i4 

4? 

4761 

5677 

6354 

6788 

6977   69i7 

66o5 

6o38 

52i4 

4128 

i3 

48 

4944 

5857 

653o 

696o 

7i45   -7081 

6764 

6i93 

5364 

4275 

12 

49 

5i28 

6o37 

6706 

7132 

73i3'  7244 

6924 

6348 

55i5 

442i 

II 

5o 

7753i2 

7862i7 

796882 

807304 

8i748o  8274o7 

837o83 

8465o3 

855665 

864567 

:o 

5i 

5496 

6396 

7°57 

7475 

7648 

757i 

7242 

6658 

58i6 

47i3 

9 

52 

5679j   6576 

7233 

7647 

78i5 

7734 

74oi 

68:3 

5966 

486o 

8 

53 

5863|   6756 

7408 

7818 

7982 

7897 

756o 

6967 

6117 

5oo6 

7 

54 

6o46 

6935 

7584 

799° 

8i5o 

8060 

7719 

7122 

6267 

5i5i 

6 

55 

623o 

7ii4 

7759 

8161 

83i7 

8223 

7878 

7277 

64i7 

5297 

5 

56 

64i3 

7294 

7935 

8333 

84841   8386 

8o36 

743  1 

6567 

5443 

4 

57 

65g6 

7473 

8110 

85o4 

865i 

8549 

8i95 

7585 

6718 

5589 

3 

58 

6780   7652 

8285 

8675 

8818 

87I2 

8354 

774o 

6868 

5734 

a 

59 

6963 

7832 

846o 

8846 

8985 

8875 

85i2 

7894 

7017 

588o 

i 

39° 

38° 

37° 

36° 

35° 

34° 

33° 

32° 

31° 

30° 

d 

Natural  Co-sines. 

i 

£#3.08 

3.02 

2.95 

2.88 

2.8l    2.75 

2.68 

2.60 

2.53   a.  46 

NATURAL   TANGENTS. 


127 


d 

50° 

51° 

52° 

53° 

54° 

55° 

56° 

57° 

58° 

59° 

0 

r.  19175 

1.23490 

1.27994 

1.32704 

1.37638 

1.42815 

1.48256 

1.53986 

i.6oo33 

1.66428 

60 

I 

9246 

3563 

8071 

2785 

7722 

2903 

8349 

4o85 

oi37 

6538 

59 

2 

93i6 

3637 

8i48 

2865 

7807 

2992 

8442 

4i83 

0241 

6647 

58 

3 

9387 

3710 

8225 

2946 

7891 

3o8o 

8536 

4281 

o345 

6757 

57 

4 

9457 

3784 

83o2 

3026 

7976 

3169 

8629 

4379 

o44g 

6867 

56 

5 

9528 

3858 

8379 

3107 

8060 

3258 

8722 

4478 

o553 

6978 

55 

G 

9-99 

393i 

8456, 

3i87 

8i45 

3347 

8816 

4576 

o657 

•7088 

54 

7 

9669 

4oo5   8533 

3268 

8229 

3436 

8909 

4675 

o76i 

7198 

53 

8 

9?4o 

4079 

8610 

3349 

83i4 

3525 

9003 

4774 

o865 

73o9 

52 

9 

9811 

4i53 

8687 

343o 

8399 

36i4 

9097 

4873 

o97o 

74i9 

5i 

10 

1.19882 

1.24227 

1.28764 

i.335n 

1.38484 

1.43703 

1.49190 

1.54972 

i.6io74 

i.6753o 

5o 

ii 

9953 

43oi 

8842 

3592 

8568 

•  3792 

9284 

5071 

"79 

764i 

49 

12 

1.20024 

4375 

8919 

3673 

8653 

388i 

9378 

5i7o 

1283 

7752 

48 

i3|   oog5 

4449 

8997 

3754 

8738 

397o 

9472 

5269 

1  388 

7863 

47 

i4 

0166 

4523 

9074 

3835 

8824 

4o6o 

9566 

5368 

1493 

7974 

46 

i5 

0237 

4597 

9152 

3916 

89o9 

4i4g 

9661 

5467 

i598 

8o85 

45 

16 

o3o8 

4672 

9229 

3998 

8994 

4239 

9755 

5567 

I7o3 

8196 

44 

l? 

°379 

4746 

93°7 

4079 

9079 

4329 

9849 

5666 

1808 

83o8 

43 

18 

o45  1 

4820 

9385 

4i6o 

9i65 

44i8 

9944 

5766 

1914 

8419 

42 

*9 

0622 

4895 

9463 

4242 

9250 

45o8 

i.5oo38 

5866 

2019 

853i 

4i 

20 

1.20593 

1.24969 

1.29541 

1.34323 

i.39336 

1.44598 

i.5oi33 

i.55966 

1.62125 

1.68643 

4o 

21 

o665 

5o44 

9618 

44o5 

9421 

4688 

0228 

6o65 

223O 

8754 

39 

22 

0736 

5n8 

9696 

4487 

95o7 

4778 

O322 

6i65 

2336 

8866 

38 

23 

0808 

5i93 

9775 

4568 

9593 

4868 

0417 

6265 

2442 

8979 

37 

24 

0879 

5268 

9853 

465o 

9679 

4958 

o5i2 

6366 

2548 

9091 

36 

25 

ogS  i 

5343 

993i 

4732 

9764 

5o49 

0607 

6466 

2654 

9203 

35 

26 

1023 

54i? 

i.Sooog 

48i4 

985o 

5i39 

0702 

6566 

276o 

93i6 

34 

27 

1094 

5492 

0087 

4896 

9936 

5229 

0797 

6667 

2866 

9428 

33 

28 

1166 

5567 

0166 

4978 

1.40022 

5320 

0893 

6767 

29-72 

954i 

32 

29 

1238 

5642 

0244 

5o6o 

oio9 

54io 

0988 

6868 

3o79 

9653 

3i 

3o 

i  aiSio 

1.25717 

i.3o323 

i.35i4a 

1.40195 

i.455oi 

i.5io84 

1.56969 

i.63i85 

1.69766 

3o 

3i 

i382 

5792 

64oi 

5224 

028: 

5592 

1179 

7069 

3292 

9879 

29 

32 

i454 

5867 

o48o 

53o7 

0367 

5682 

1275 

7170 

3398 

9992 

28 

33 

i526 

5943 

o558 

5389 

o454 

5773 

1370 

7271 

35o5 

i.7oio6 

27 

34 

i598 

6018 

0637 

5472 

o54o 

5864 

i466 

7372 

36i2 

0219 

26 

35 

1670 

6o93 

0716 

5554 

0627 

5955 

i562 

7474 

37i9 

o332 

25  i 

36 

1742 

6169 

0795 

5637 

0714 

6o46 

i658 

7575 

3826 

o446 

24 

3? 

1814 

6244 

0873 

57i9 

0800 

6137 

1754 

7676 

3934 

o56o 

23 

38 

1886 

63i9 

0952 

58o2 

0887 

6229 

i85o 

7778 

4o4i 

o673 

22 

39 

1959 

6395 

io3i 

5885 

o974 

6320 

1946 

7879 

4i48 

o787 

21 

4o 

I.22O3I 

1.26471 

i.Snio 

1.36968 

1.41061 

i.464ii 

1.52043 

i.5798i 

1.64256 

1.70901 

20 

4i 

2IO4 

6546 

1190 

6o5i 

n48 

65o3 

2i3g 

8o83 

4363 

ioi5 

J9 

42 

2176 

6622 

1269 

6i34 

1235 

6595 

2235 

8i84 

447i 

1129 

18 

43 

2249 

6698 

1  348 

6217 

1322 

6686 

2332 

8286 

4579 

1244 

17 

44 

2321 

6774 

1427 

63oo 

i4o9 

6778 

2429 

8388 

4687 

i358 

16 

45 

23g4 

6849 

1507 

6383 

1497 

6870 

2525 

8490 

4795 

i473 

i5 

46 

2467 

6925 

i586 

6466 

i584 

6962 

2622 

8593 

4903 

i588 

i4 

4? 

2539 

7001 

1666 

6549 

1672 

7o53 

2719 

8695 

Son 

1702 

i3 

48 

2612 

7077 

1745 

6633 

i759 

7146 

2816 

8797 

5l2O 

1817 

12 

49 

2685 

7i53 

1825 

6716 

1  847 

7238 

2913 

8900 

52a8 

I932 

II 

5o 

1.22758 

1.27230 

1.31904 

i.36Soo 

1.41934 

i.4733o 

i.53oio 

1.59002 

i.65337 

1.72047 

10 

5i 

283i 

73o6 

1984 

6883 

2O22 

7422 

3107 

9io5 

5445 

2i63 

9 

52 

2904 

7382 

2064 

6967 

2IIO 

75i4 

32o5 

9208 

5554 

2278 

8 

53 

2977 

7458 

2144 

7o5o 

2198 

7607 

33o2 

93ii 

5663 

2393 

7 

54 

3o5o 

7535 

2224 

7i34 

2286 

7699 

34oo 

94i4 

5772 

25o9 

6 

55 

3:23 

7611 

23o4 

7218 

a374 

7792 

3497 

95i7 

588i 

2625 

5 

56 

3196 

7688 

2384 

7302 

2462 

7885 

3595 

9620 

599o 

2741 

4 

5? 

3270 

7764 

2464 

7386 

255o 

7977 

3693 

9723 

6o99 

2857 

3 

58 

3343 

784i 

2544 

7470 

2638 

8070 

379i 

98261   6209 

2973 

2 

59 

34i6 

7917 

2624 

7554 

2726 

8i63 

3888 

9930   63i8 

3089 

I 

39° 

38°   37° 

36° 

35° 

34° 

33° 

32°   31° 

300  L- 

Natural  Co-tangents. 

P.  P. 
to  I".1'20 

1.26 

i.3i 

i.37 

i.44 

i.5i 

i.59 

1.68   1.78 

1.88 

128 


NATURAL    SINES. 


j 

60°  |  61° 

62° 

63° 

64° 

65° 

66° 

67° 

68°  |  69° 

0 

866025 

874620 

882948 

891007 

898794 

9o63o8 

9i3545 

92o5o5 

0.27184!  93358o!6o 

6171 

4761 

3o84 

n39 

8922 

643i 

3664 

0618 

7293|   3685 

59 

3 

63i6 

4902 

3221 

1270 

9049 

6554 

3782 

0732 

7402 

3789 

58 

j 

646i 

5o42 

3357 

1402 

9176 

6676 

39oo 

o846 

75io 

3893 

57 

; 

6607 

5i83 

3493 

i534 

93o4 

6799 

4oi8 

o959 

7619 

3997 

56 

1 

6752 

5324 

3629 

1666 

943i 

6922 

4i36 

IO72 

7728 

4ioi 

55 

6 

6897 

5465 

3766 

1798 

9558 

7044 

4254 

n85 

7836 

4204 

54 

7 

7042 

56o5 

3902 

1929 

c;685 

7166 

4372 

i299 

7945 

43o8 

53 

8 

7187 

5746 

4o38 

2061 

9812 

7280. 

449o 

1412 

8o53 

44i2 

52 

9 

733i 

5886 

4174 

2192 

9939 

74*1 

46o7 

i525 

8161 

45i5 

5i 

10 

867476 

876026 

884309 

892323 

900065 

9o7533 

9i4725 

92i638 

928270 

9346i9 

5o 

ii 

7621 

6167 

4445 

2455 

0192 

7655 

4842 

i75o 

8378 

4722 

49 

12 

7765 

63o7 

458i 

2586 

0319 

7777 

496o 

1  863 

8486 

4826 

48 

i3 

7910 

6447 

4717 

2717 

o445 

7899 

5o77 

i976 

8594 

4929 

47 

i4 

8o54 

6587 

4852 

2848 

0572 

8021 

5i94 

2088 

8702 

5o32 

46 

i5 

8199 

6727 

4988 

2979 

0698 

8i43 

53ii 

22OI 

8810 

5i35 

45 

16 

8343 

6867 

5i23 

3no 

o825 

8265 

5429 

23i3 

8917 

5238 

44 

17 

8487 

7006 

5258 

324i 

o95i 

8387 

5546 

2426 

9025 

534i 

43 

18 

8632 

7i46 

5394 

337i 

1077 

85o8 

5663 

2538 

9i33 

5444 

42 

19 

8776 

7286 

5529 

35o2 

1203 

863o 

5779 

265o 

9240 

5547 

4i 

20 

868920 

877425 

885664 

893633 

901329 

9o875i 

9i5896 

0,22762 

929348 

93565o 

4o 

21 

9064 

7565 

5799 

3763 

i455 

8872 

6oi3 

2875|   9455 

5752 

39 

22 

9207 

77o4 

5934 

3894 

i58i 

8994 

6i3o 

2986 

9562 

5855 

38 

23 

935i 

7844 

6069 

402^ 

1707 

9n5 

6246 

3o98 

9669 

5957 

37 

24 

9495 

7983 

6204 

4i54 

i833 

9236 

6363 

3210 

9776 

6060 

36 

£ 

9639 

8122 

6338 

4284 

1958 

9357 

6479 

3322 

9884 

6162 

35 

26 

9782 

8261 

6473 

44i5 

2084 

9478 

6595 

3434 

999° 

6264 

34 

27 

9926 

84oo 

6608 

4545 

2209 

9599 

67I2 

3545 

93oo97 

6366 

33 

28 

870069 

8539 

6742 

4675 

2335 

9720 

6828 

3657 

0204 

6468 

32 

29 

O2  I  2 

8678 

6876 

48o5 

2460 

984i 

6944 

3768 

o3n 

657o 

3i 

3o 

870356 

878817 

887011 

894934 

902585 

9o996i 

9i7o6o 

923880 

93o4i8 

936672 

3o 

3i 

0499 

8956 

7i45 

5o64 

2710 

9ioo82 

7i76 

399i 

o524 

6774 

29 

32 

O642 

9095 

7279 

5i94 

2836 

O2O2 

7292 

4lO2 

o63i 

6876)28 

33 

0785 

9233 

74i3 

5323 

2961 

o323 

7408 

4213 

o737 

6977 

27 

34 

0928 

9372 

7548 

5453 

3o86 

o443 

7523 

4324 

oS43 

7079 

26 

35 

1071 

95io 

7681 

5582 

3210 

o563 

7639 

4435 

o95o 

7181 

25 

36 

I2l4 

9649 

78i5 

5712 

3335 

0684 

7755 

4546 

io56 

7282 

24 

3y 

i357 

9787 

7949 

584i 

346o 

o8o4 

7870 

4657 

1162 

7383 

23 

38 

1499 

9925 

8o83 

5970 

3585 

0924 

7986 

4768 

1268 

7485 

22 

39 

1642 

88oo63 

8217 

6099 

3709 

io44 

8101 

4878 

i374 

7586 

21 

4o 

871784 

880201 

88835o 

896229 

903834 

9iii64 

9i82i6 

924989 

93i48o 

937687 

2O 

4i 

1927 

oSSg 

8484 

6358 

3958 

1284 

833i 

5099 

i586 

7788 

'9 

42 

2069 

o477 

8617 

6486 

4o83 

i4o3 

8446 

52IO 

i69i 

7889 

18 

43 

2212 

o6i5 

875i 

66i5 

4207 

i523 

856i 

5320 

1797 

799° 

ll 

44 

2354 

0753 

8884 

6744 

433i 

i643 

8676 

543o 

I9O2 

8o9i 

16 

45 

2496 

0891 

9017 

6873 

4455 

1762 

879i 

554i 

2008 

8i9i 

i5 

46 

2638 

1028 

9i5o 

7001 

4579 

1881 

89o6 

565i 

2Il3 

8292 

i4 

4? 

2780 

1166 

9283 

7i3o 

47o3 

2OOI 

9O2I 

6761 

22I9 

8393 

i3 

48 

2922 

i3o3 

94i6 

7258 

482? 

2120 

9i35 

587i 

2324 

8493 

12 

49 

3o64 

i44i 

9549 

7387 

495i 

2239 

925o 

5980 

2429 

8593 

II 

5o 

873206 

88i578 

889682 

8975i5 

9o5o75 

9I2358 

9i9364 

926090 

932534 

938694 

IO 

5i 

3347 

1716 

98i5 

7643 

5i98 

2477 

9479 

6200 

2639 

8794 

9 

52 

3489 

i853 

9948 

7771 

5322 

2596 

9593 

63io 

2744 

8894 

8 

53 

363i 

1990 

890080 

7900 

5445 

2715 

9707 

6419 

2849 

8994 

7 

54 

3772 

2127 

02l3 

8028 

5569 

2834 

982I 

6529 

2954 

9o94 

6 

55 

3914 

2264 

o345 

8:56 

5692 

2953 

9936 

6638 

3o58 

9i94 

5 

56 

4o55 

2401 

0478 

8283 

58i5 

3072 

92oo5o 

6747 

3i63 

9294 

4 

57 

4196 

2538 

0610 

84n 

5939 

3i9o 

oi64l   6857 

3267 

c;394 

3 

58 

4338 

2674 

0742 

8539 

6062 

33o9 

0277 

6966 

3372 

9493 

2 

59 

4479 

2811 

o874 

8666 

6i85 

3427 

o39i 

7o75 

3476 

9593 

I 

29°  |  28°   27° 

26° 

25° 

24°   23° 

22°   21° 

20° 

d 

Natural  Co-sines. 

9 

££>39|  2.3!  |   2.24 

2.16 

2.O9 

2.01 

i.93 

1.86 

1.78 

1.70 

NATURAL    TANGENTS. 


129 


J 

s 

60° 

61° 

62° 

63° 

64° 

65° 

66° 

67° 

G8° 

69° 

— 

o 

1.73205 

i.8o4o5 

i.88o73 

1.96261 

a.o5o3o 

a.i445i 

2.2460^ 

2.35585 

2.47509 

2.6o5o9 

60 

I 

332i 

o52g 

8205 

6402 

5i82 

46i4 

478o 

5776 

7716 

0736 

59 

2 

3438 

o653 

8337 

6544 

5333 

4777 

4956 

5967 

7924 

o963 

58 

r 

3555 

°777 

8469 

6685 

5485 

494o 

5i32 

6i58 

8x32 

II9O 

57 

j! 

367i 

0901 

8602 

6827 

5637 

5io4 

53o9 

6349 

834o 

i4i8 

56 

t 

3788 

1025 

8734 

6969 

579o 

5268 

5486 

654i 

8549 

1  646 

55 

6 

SgoS 

u5o 

8867 

•7111 

5942 

5432 

5663 

6733 

8758 

1874 

54 

7 

4O22 

127^ 

9ooo 

7253 

609^ 

5596 

584o 

6920 

8967 

2:o3 

53 

8 

4i4o 

i399 

9i33 

7395 

6247 

5760 

6018 

7n8 

9177 

2332 

52 

9 

4s57 

i524 

9266 

7538 

64oo 

5925 

6196 

73u 

9386 

256i 

5i 

10 

!.74375 

1.81649 

i.894oo 

i.9768i 

2.o6553 

2.j6o9o 

2.26374 

2.375o4 

2.49697 

2.6279I 

5o 

ii 

4492 

i774 

9533 

7823 

67o6 

6255 

6552 

7697 

9807 

3021 

49 

12 

46io 

1899 

9667 

7966 

6860 

6420 

673o 

7891 

2.5ooi8 

3252 

48 

i3 

4728 

2O25 

98oi 

8110 

•701^ 

6585 

69o9 

8o84 

0229 

3483 

47 

i4 

4846 

2i5o 

9935 

8253 

7167 

675i 

7088 

8279 

o44o 

37i4 

46 

i5 

4964 

2276 

i.9oo69 

8396 

732I 

69i7 

7267 

8473 

o652 

3945 

45 

16 

5o82 

24O2 

O2O3 

854o 

7476 

7083 

7447 

8668 

0864 

4i77 

44 

17 

6200 

2528 

o337 

8684 

763o 

7249 

7626 

8863 

1076 

44io 

43 

18 

5319 

2654 

o472 

8828 

7785 

74i6 

7806 

9o58 

I289 

4642 

42 

J9 

5437 

278o 

o6o7 

8972 

7939 

7582 

7987 

9253 

i5o2 

4875 

4i 

20 

i.75556 

1.82906 

i.9o74i 

i.99n6 

2.08092! 

2-I7749 

2.28167 

2.39449 

2.51715 

2.65io9 

4o 

21 

5675 

3o33 

o876 

9261 

825o 

79i6 

8348 

9645 

I929 

5342 

39 

22 

5794 

SiSg 

IOI2 

9406 

84o5 

8o84 

8528 

9841 

2142 

5576 

38 

23 

59i3 

3286 

n47 

955o 

856o 

825i 

8710 

2.4oo38 

2357 

58ii 

37 

24 

6o32 

34i3 

1282 

9695 

8716 

84i9 

889i 

0235 

2571 

6o46 

36 

25 

6i5i 

354o 

i4i8 

984i 

8872 

8587 

9o73 

0432 

2786 

6281 

35 

26 

627I 

3667 

i554 

9986 

9028 

8755 

9254 

0629 

3ooi 

65i6 

34 

27 

6890 

3794 

i69o 

2.00l3l 

9184 

8923 

9437 

o827 

3217 

6752 

33 

28 

65io 

3922 

1826 

0277 

934i 

9o92 

96i9 

1025 

3432 

6989 

3i 

29 

663o 

4o49 

I962 

o423 

9498 

926l 

98oi 

1223 

3648 

7225 

3x 

3o 

1.76749 

i.84i77 

I.92O98 

2.00669 

2.09654 

2.i943o 

2.29984 

2.41421 

2.53865 

2.67462 

3o 

3i 

6869 

43o5 

2235 

o7i5 

9811 

9599 

2.3oi67 

l620 

4082 

7700 

29 

32 

6990 

4433 

237I 

0862 

9969 

9769 

o35i 

1819 

4299 

7937 

28 

33 

7no 

456i 

25o8 

1008 

2.10126 

9938 

o534 

2019!  45i6 

8i75 

27 

34 

723o 

4689 

2645 

n55 

0284 

2.20108 

0718 

2218 

4734 

84i4 

26 

35 

735i 

48i8 

2782 

1302 

0442 

0278 

09O2 

2418 

4952 

8653 

25 

36 

747i 

4946 

2920 

1  449 

0600 

o449 

1086 

2618 

5170 

8892 

24 

3? 

7592 

5o75 

3o57 

i596 

o758 

o6i9 

I27I 

2819 

5389 

9i3i 

23 

38 

77i3 

6204 

3i95 

I743 

0916 

0790 

i456 

3019 

56o8 

937i 

22 

39 

7834 

5333 

3332 

1891 

io75 

0961 

i64i 

322O 

5827 

96l2 

21 

4o 

1.77966 

1.85462 

i.9347o 

2.O2O39 

2.II233 

2.2II32 

2.31826 

2.43422 

2.56o46 

2.69853 

20 

4i 

8o77 

559i 

36o8 

2l87 

1392 

:3o4 

2OI2 

3623 

6266 

2.7O09/! 

J9 

42 

8198 

5720 

3746 

2335 

i552 

1475 

2I97 

3825 

648  7 

o335 

18 

43 

8319 

585o 

3885 

2483 

i7n 

1  647 

2383 

4O27 

6707 

o577 

!? 

44 

844i 

5979 

4oa3 

263i 

1871 

1819 

257O 

4230 

6928 

o8i9 

16 

45 

8563 

6io9 

4162 

278o 

2o3o 

1992 

2756 

4433 

7i5o 

1062 

i5 

46 

8685 

6239 

43oi 

2929 

2190 

2164 

2943 

4636 

737i 

i3o5 

i4 

4? 

88o7 

6369 

444o 

3o78 

235o 

2337 

3i3o 

4839 

7593 

i548 

i3 

48 

8929 

6499 

4579 

3227 

25l  I 

25lO 

33i7 

5o43 

78i5 

I792 

12 

49 

goSi 

663o 

47i8 

3376 

2671 

2683 

35o5 

5246 

8o38 

2o36 

II 

5o 

1.79174 

i.8676o 

i.94858 

2.o3526 

2.12832 

2.22857 

2.33693 

2.4545i 

2.58261 

2.72281 

IO 

5i 

9296 

689i 

4997 

3675 

2993 

3o3o 

388i 

5655 

8484 

2526 

g 

52 

9419 

•7021 

5i37 

3825 

3i54 

32o4 

4o69 

586o 

8708 

2771 

8 

53 

9542 

7l52 

5277 

3975 

33i6 

3378 

4258 

6o65 

8932 

3017 

7 

54 

9665 

7283 

54i7 

4i25 

3477 

3553 

4447 

627O 

9i56 

3263 

6 

55 

9788 

74i5 

5557 

4276 

3639 

3727 

4636 

6476 

938i 

35o9 

5 

56 

9911 

7546 

5698 

4426 

38oi 

3902 

4825 

6682 

96o6 

3756 

4 

5? 

i.8oo34 

7677 

5838 

4577 

3963 

4077 

5oi5 

6888 

983i 

4oo4 

3 

58 

oi58 

78o9 

5979 

4728 

4i25 

4252 

52o5 

7095 

2.60057 

4261 

2 

69 

0281 

794i 

6120 

4879 

4288 

4428 

5395 

7302 

0283 

4499 

I 

29° 

28° 

27° 

26° 

25° 

24° 

23° 

22° 

21° 

20° 

a 

Natural  Co-tangents. 

i 

P.  P 

fol''.2'00 

2.13 

2.27 

2.44 

2.62 

2.82 

3.o5 

3.3i 

3.6i 

3.95 

j 

130 


NATURAL 


s 

s 

70° 

71° 

72° 

73° 

74° 

75° 

7G> 

77° 

78° 

79° 

o 

939693 

9455i9 

95io57 

9563o5 

96l2&2 

965926 

970296 

97437o 

978i48 

98162-7 

5o 

I 

9792 

56i3 

n46 

6390 

I  342 

6001 

o366 

4435 

8208 

1  683 

5o 

2 

9891 

57o8 

1236 

6475 

l422 

6o76 

o436 

45oi 

8268 

i738 

58 

3 

9991 

58o2 

i326 

65  60 

1502 

6i5i 

o5o6 

4566 

8^29 

i793 

57 

4 

940090 

5897 

i4i5 

6644 

i582 

6226 

o577 

463i 

8389 

i849 

56 

5 

0189 

599i 

i5o5   6729 

1662 

63oi 

0647 

4696 

8449 

i9o4 

55 

6 

0288 

6o85 

i594   68i4 

i74i 

6376 

0716 

476i 

85o9 

i959 

54 

7   o387 

6180 

1684:   6898 

1821 

645  1 

0786 

4826 

8569 

20l4 

53 

8   o486 

6274 

i773|   6983 

I9OI 

6526 

o856 

489i 

8629 

2o69 

52 

9 

o585 

6368 

1862   7o67 

I98o 

6600 

0926 

4956 

8689 

2123 

5i 

10 

940684 

946462 

95i95i 

957i5i 

962o59 

966675 

97°995 

97502O 

978748 

982i78 

5o 

ii 

0782 

6555 

2o4o|   7235 

2l39 

6749 

io65 

5o85 

8808 

2233 

49 

12 

0881 

6649 

2I29 

7319 

2218 

6823 

n34 

5i49 

886  7 

2287 

48 

i3 

0979 

6743 

2218 

74o4 

2297 

6898 

1204 

5214 

892-7 

2342 

47 

i4 

1078 

6837 

23o7   7487 

2376 

6972 

I273 

5278 

8986 

2396 

46 

i5 

1176 

6930 

2396;   7571 

2455 

7046 

1  342 

5342 

9045 

245o 

45 

16 

1274 

7024 

2484^   7655 

2534 

7120 

i4n 

54o6 

9io5 

25o5 

<4 

*7 

1372 

7ii'7 

2573 

7739 

26i3 

7194 

i48o 

547i 

9164 

2559 

43 

18 

i4yi 

7210 

2661 

7822 

2692 

7268 

1  549 

5535 

9223 

26i3 

4s 

J9 

1669 

73o4 

2750 

7906 

2770 

7342 

1618 

5598!  9282 

2667 

4i 

20 

o4i666 

947397 

952838 

957990 

962849 

967415 

97i687 

975662 

97934i 

982721 

4o 

21 

1764 

7490 

2926 

8o73 

2928 

7489 

i755 

5726 

9399 

2774 

39 

22 

1862 

7583 

3oi5 

8i56 

3oo6 

7562 

1824 

579o 

9458 

2828 

38 

23 

1960 

7676 

3io3 

8239 

3o84 

7636 

i893 

5853 

95i7 

2882 

37 

24 

2057 

7768 

3191   8323 

3i63 

7709 

1961 

59i7 

9575 

2935 

36 

25 

2i55 

7861 

3279 

84o6 

324l 

7782 

2029 

598o 

9634 

2989 

35 

26 

2252 

7954 

3366 

8489 

33i9 

7856 

2098 

6o44 

9692 

3o42 

34 

27 

235o 

8o46 

3454 

8572 

3397 

7929 

2166 

6107 

975o 

3o96 

33 

«8 

2447 

8139 

3542 

8654 

3475 

8002 

2234 

6170 

98o9 

3i49 

32 

29 

2544 

823i 

3629 

8737 

3553 

8o75 

23O2 

6233 

9867 

32O2 

3i 

3o 

942641 

948324 

9537i7 

958820 

96363o 

968148 

97237o 

976296 

979925 

983255 

3o 

3i 

2739 

84i6 

38o4 

8902 

37o8 

8220 

2438 

6359 

9983 

33o8 

29 

32 

2836 

85o8 

3892 

8985 

3786 

8293 

25o6 

6422 

98oo4i 

336i 

28 

33 

2932 

8600 

3979 

906? 

3863 

8366 

2573 

6485 

oo98 

34i4 

27 

34   3o2o 

8692 

4o66 

9i5o 

394i 

8438 

2641 

6547 

oi56 

3466 

26 

35 

3i26 

8784 

4i53 

9232 

4oi8 

85n 

27o8 

6610 

O2l4 

35i9 

25 

36 

3223 

8876 

4s4o 

93i4 

4095 

8583 

2776 

6672 

027I 

357i 

24 

37   33i9 

8968 

4327 

9396 

4i73 

8656 

2843 

6735 

o329 

3624 

23 

38   34i6 

9o59 

44i4 

9478 

'  425o 

8728 

291  1 

6797 

o386 

3676 

22 

39'   35i2 

9i5i 

45oi 

956o 

4-327 

8800 

2978 

6859 

o443 

3729 

21 

4o 

943609 

949243 

954588 

959642 

964404 

968872 

973o45 

97692i 

98o5oo 

98378i 

2O 

4.i 

37o5 

9334 

4674 

9724 

448  1 

8944 

3lI2 

6984 

o558 

3833 

X9 

42 

38oi 

94a5 

4761 

98o5 

4557 

9016 

3i79 

7o46 

o6i5 

3885 

18 

43 

3897 

95i7 

4847 

9887 

4634 

9088 

3246 

7108 

o672 

3937 

17 

44 

3993 

9608 

4934 

9968 

4711 

9l59 

33i3 

7i69 

0-728 

3989 

16 

45 

4o89 

9699 

5O2O 

96oo5o 

4787 

9231 

3379 

7231 

o785 

4o4i 

i5 

46 

4i85 

9790 

5io6 

oi3i 

4864 

93o2 

3446 

7293 

0842 

4O92 

i4 

47 

4281 

9881 

5l92 

O2  I  2 

4940 

9374 

35i2 

7354 

o899 

4i44 

i3 

43 

4376 

9972 

5278 

O294 

5oi6 

9445 

3579 

74i6 

o955 

4i96 

12 

49 

4472 

95oo63 

5364 

o375 

5o93 

95l7 

3645 

7477 

1012 

4247 

II 

5o 

944568 

95oi54 

95545o 

96o456 

965169 

969588 

9737i2 

977539 

98io68 

984298 

10 

5i 

4663 

0244 

5536 

o537 

5245 

9659 

3778 

7600 

1124 

435o 

9 

52 

4?58 

o335 

5622 

0618 

532i 

973o 

3844 

7661 

1181 

44oi 

8 

53 

4854 

o425 

57o7 

o698 

5397 

98oi 

39ro 

7722 

I237 

4452 

7 

54 

4949 

o5i6 

5793 

°779 

5473 

9872 

3976 

7783 

I293 

45o3 

6 

55 

5o44 

0606 

5879 

0860 

5548   9943 

4o4s 

7844 

1  349 

4554 

5 

56 

5i39 

o696 

5964 

o94o 

5624 

970014 

4io8 

79o5 

i4o5 

46o5 

4 

5? 

5234 

0786 

6o49 

1021 

57oo 

oo84 

4173 

7966 

i46o 

4656 

3 

58 

5329 

0877 

6i34 

IIOI 

5775 

oi55 

4239 

8026 

i5i6 

47o7 

2 

_59 

542/ 

0967 

6220 

1181 

585o 

O225 

43o5 

8o87 

l572 

4757 

I 

19> 

18° 

17° 

]6° 

15°   14° 

13° 

12° 

11° 

10° 

d 

Natural  Co-sines. 

1 

£'£'•«• 

i.54 

i.46 

i.38 

:.3o 

I.  21 

i.i3 

i.o5 

0.97  '  0.88 

NATURAL    TANGENTS. 


131 


1 

70° 

71° 

72° 

73° 

74° 

75° 

76° 

77° 

78° 

79° 

o 

2.74748 

2.90421 

3.07768 

3.27o85 

3.4874i 

3.732o5 

4-oio78 

4.33i48 

4.70463 

5.14455 

60 

I 

4997 

0696 

8o73 

7426 

9126 

364o 

16-76 

3723 

1137   6266 

69 

a 

'  5246 

0971 

8379 

7767 

9609 

4o75 

2074 

43oo 

i8i3,   6068 

58 

3 

5496 

1246 

8685 

8109 

9894 

4612 

2574 

4379 

2490.   6863 

67 

4 

5746 

i523 

8991 

8452 

3«OO2t7Ql    L\.Q3Q    OO70 

5459i   3i7o|   767i 

56 

5 

5996 

1799 

9298 

8795 

0666 

5388,   3578 

6o4oi   385i 

848o 

55 

6 

6247 

2076 

9606 

9i39 

io53 

6828 

4o8i 

6623 

4534 

9293 

54 

7 

6498i   2354 

9914'  9483;  i44i 

6268 

4586 

7207 

6219 

6.20107 

53 

8 

6760 

2632 

3.IO223   9829 

1829 

67o9 

5o92 

7793 

6906 

0926 

62 

9 

7002 

2910 

o532 

3.30174 

2219 

7162 

5599 

838i 

6595 

1744 

61 

10 

2.77254 

2.93i89'3.io842 

3.3o52i 

3.62609 

3.77595 

4.0610-7 

4.38969 

4.77286 

5.22666 

5o 

ii 

75°7 

3468 

n53 

0868 

3ooi 

8o4o 

6616 

956o 

7978 

339i 

49 

12 

7761 

3748 

i464 

1216 

3393 

8485 

7127 

4.40162 

8673 

4218 

48 

i3 

8oi4 

4028 

i775 

i565 

3785 

893i 

7639 

o745 

937o 

5o48 

47 

i4 

8269 

43o9 

2o87 

1914 

4179 

9378|   8162 

i34o 

4.80068 

6880 

46 

i5 

8523 

459i 

2400 

2264 

4573 

9827 

8666 

I936 

o769 

6716 

45 

16 

8778 

4872 

27l3|    26l4 

4968 

3.8o276 

9l82 

2534 

i47i 

7553 

44 

17 

9o33 

5i55 

3027 

2965 

5364 

0-726 

9699 

3i34 

2I75 

8393 

43 

18 

9289 

5437 

334i 

33i7 

576i 

n77 

4.10216 

3735 

2882 

9235 

42 

19 

9545 

6721 

3656 

367o 

6169 

i63o 

0736 

4338 

3690 

5.3oo8o 

4i 

20 

2«798O2 

2.96004 

3.i3972 

3.34023 

3.56557 

3.82083 

4.ii256'444942 

4-84300 

6.30928 

4o 

21 

2.80060, 

6288 

4288 

4377 

696-7 

2537 

1778 

5548 

5oi3 

1778 

39 

22 

o3i6 

6573 

46o5 

4732 

7357 

2992 

2301 

6i55 

57?7 

263i 

38 

23 

o574 

6858 

4922 

5o87 

7758 

3449 

2826 

6764 

6444 

3487 

37 

24 

o833j   7144 

524o 

5443 

8160 

39o6 

335o 

7374 

7162 

4345 

36 

25 

io9i 

743o 

5558 

58oo 

8662 

4364 

3877 

7986 

7882 

6206 

35 

26 

i35o 

77i7 

5877 

6i58 

8966 

4824 

44o5 

8600 

8606 

6070 

34 

27 

1610 

8oo4 

6i97l   65i6l   937o 

6284 

4934 

92l5 

933o 

6936 

33 

28 

1870 

8292 

65i7 

6875   9775 

5745 

5465 

0832 

4.90066 

7806 

32 

29 

2i3o 

858o 

6838 

7234'3.  60181 

6208 

5997 

4.5o45i 

0786 

8677 

3i 

3o 

2.8239I 

2.98868 

3.i7i59 

3.375943.6o588 

3.8667i 

4.i653o 

4.61071 

4.91616 

6.39662 

3o 

3t 

2653 

9  1  58)   748i 

7955   0996 

7i36 

7064 

i693 

2249 

5.40429 

29 

32 

29l4 

94471   7804 

83i7   i4o5 

76oi 

7600 

23i6 

2984 

1  309 

28 

33 

3176 

9738 

8127:   8679,   1814 

8068 

8i37 

294i 

3721 

2192 

27 

34 

3439 

3.00028 

845il   9042 

2224 

8536 

8676 

3568 

446o 

3077 

26 

35 

3702 

oSig 

8775!   9406 

2636 

9oo4 

92l5 

4i96 

6201 

3966 

25 

36 

3965 

06  1  1 

9100 

9771 

3o48 

9474 

9766 

4826 

5945 

4857 

24 

37 

4229 

0903 

9426 

3.4oi36 

346i 

9945 

4.2020,8 

5458 

6690 

6761 

23 

38 

4494 

1196 

9752 

o5o2 

3874 

3.9o4i7 

0842 

6o9i 

7438 

6648 

22 

39 

4768 

1489 

3.200-79 

0869 

4289 

0890 

i387 

6726 

8188 

7548 

21 

4o 

2.85o23 

3.oi783 

3.2o4o6 

3.4i236 

3.647o5 

3.9i364 

4.2i933 

4.57363 

4-9894o 

5.4845i 

2O 

4i 

5289 

2077 

o734 

i6o4 

6121 

i839 

2481 

8001 

9696 

9356 

'9 

42 

5555 

2372 

io63 

I973 

5538 

23i6 

3o3o 

864i 

5.  0046  1 

6.60264 

18 

43 

5822 

2667 

1392 

2343 

6967 

2793 

358o 

9283  -   1210 

1176 

17 

44 

6089 

2963 

I722 

2713 

6376 

327I 

4l32 

9927 

1971 

2090 

16 

45 

6356 

3260 

2o53 

3o84 

6796 

375i 

4685 

4.60672 

2734 

3007 

16 

46 

6624 

3556 

2384 

3456 

7217 

4232 

5239 

1219 

3499 

3927 

i4 

47 

6892 

3854 

27l5 

3829 

7638 

47i3 

5795 

1868 

4267 

485  1 

i3 

48 

7161 

4i52 

3o48 

4202 

8061 

6196 

6352 

2618 

6037 

6777 

12 

49 

743o 

445o 

338i 

4576 

8485 

568o 

69n 

3i7i 

6809 

6706 

II 

5o 

2.87700 

3.o4749 

3.237i4 

3.44961 

3.68909 

3.96166 

4.27471 

4.63825 

5.06584 

5.57638 

IO 

5i 

7970 

6049 

4049 

5327 

9335 

6661 

8o32 

448o 

736o 

8573 

9 

52 

8240 

5349 

4383 

57o3 

976i 

7i39 

8595 

5i38 

8139 

9611 

8 

53 

85n 

5649 

4-7I91   6080 

3.  -70188 

7627 

9169 

5797 

8921 

6.60462 

7 

54 

8783 

595o 

5o55i   6458 

0616 

8n7 

9724 

6458 

9704 

i397 

6 

55 

9o55 

6262 

5392 

6837 

io46 

860-7 

4«3o29i 

7121  6.10490 

a344 

5 

56 

9327 

6554 

5-729 

7216 

1476 

9°99 

0860 

7786   1279 

3295 

4 

57 

9600 

6857 

6o67 

7696 

i9°7 

9592 

i43o 

8462   20691  4s48 

3 

58 

9873 

7160 

64o6 

7977 

2338  4.00086 

200  1 

9121 

2862   52o5 

2 

59 

2.90147 

7464 

6745 

8359 

2771   0682 

2573 

9791 

3658   6166 

I 

19° 

18° 

17° 

16°   15°   14° 

13°   12°   11°  i  10° 

a 

Natural  Co-tangents. 

i 

P.  P.  ,  or 

to  I".4'35 

4.8a 

5.36 

6.01 

6.79 

7.73 

8.9o  1  10.35 

12.20  14.60 

NATURAL    SINEH. 


0T 

80°  ]  81°   82° 

83° 

84° 

85° 

86° 

87° 

88° 

89° 

oig848o8  987688 

99o268 

992546 

994522 

996195 

9-97564 

99863ol99939i 

999848 

60 

i 

4858 

7734 

o3o9 

2582 

4552 

6220 

7584 

8645 

94oi 

9853 

59 

2 

4go9 

7779 

o349 

2617 

4583 

6245 

7604 

8660 

94n 

9358 

58 

3 

4959 

7824 

o389 

2652 

46i3 

6270 

7625]   8675 

942I 

9863 

57 

4 

Soog 

7870 

0429 

2687 

4643 

6295 

76451   869o 

943i 

9867 

56 

5 

5o59 

79i5 

o469 

2722 

4673 

6320 

7664)   8705 

944i 

9872 

55 

6 

5109 

796o 

o5o9 

2757 

4703 

6345 

7684,  8710. 

945o 

9877 

54 

7 

5i59 

8oo5 

o549 

2792 

4733 

637o 

7704 

8734 

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53 

8 

5209 

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2827 

4762 

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9469 

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9 

5259 

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2862 

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6419 

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10 

985309 

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99o669 

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54o8 

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6493 

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5027 

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18 

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19 

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20 

985801 

988582 

991061 

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9989i7 

999577 

999932 

4o 

21 

585o 

8626,   noo 

3272 

5i42 

6709 

7972 

893i 

9585 

9936 

39 

22 

5899 

8669|   n38;   33o6 

5170 

6732 

799° 

8944 

9594 

9939 

38 

23 

5947 

8713.   1177 

3339 

5i99 

6756 

8008 

8957 

9602 

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24 

5996 

87561   1216 

3373 

5227 

6779 

8027 

8971 

9610 

9945 

36 

25 

6o45 

8800 

1254;  3406 

5256 

6802 

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8984 

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35 

26 

6o93 

8843 

1292 

3439 

5284 

6825 

8o63 

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9626 

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34 

27 

6i4i 

8886 

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3473 

53i2 

6848 

8081 

9010 

9634 

9954 

33 

28 

6i89 

893o 

1  369 

35o6 

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6872 

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9023 

9642 

9957 

32 

29 

6238 

8973 

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3539 

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32 

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3638 

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33 

6429 

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5479 

6985 

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27 

34 

6477 

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1596 

3703 

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9098 

9687 

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26 

35 

6525 

923o   i634 

3735 

5535 

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9111 

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25 

36 

6572 

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3768 

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9976 

24 

37 

6620 

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23 

38 

6667 

9.357   1746 

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22 

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6714 

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9159 

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21 

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9736 

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42 

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9526 

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5725 

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9i94 

9743 

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1  8 

43 

69o3 

9568 

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3993 

5752 

7207 

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92o6 

9749 

9988 

17 

44 

695o 

9610 

i968 

4o25 

5778 

7229 

8375 

92l8 

9756 

9989 

16 

45 

6996 

965i 

2OO5 

4o56 

58o5 

725o 

8392 

9229 

9762 

999° 

i5 

46 

9693 

2O42 

4o88 

5832 

7272 

84o8 

924o 

9768 

9992 

i4 

47 

7090 

9735 

2078 

4l2O 

5858 

7293 

8425 

9252 

9775 

9993 

i3 

48 

7i36 

9776 

2Il5 

4i5i 

5884 

7312 

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9263 

9781 

9994 

12 

49 

7i83 

98i8 

2l5l 

4182 

5911 

7336 

8457 

927^ 

9787 

9995 

II 

5o 

987229 

989859 

992i87 

994214 

995937 

997357 

998473 

999285 

999793 

999996 

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7275 

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222^ 

4245 

5963 

7378 

8489 

9296 

9799 

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9 

52 

7322 

9942 

2260 

4276 

5989 

7399 

85o5 

980^ 

9997 

8 

53 

7368 

9983 

2296 

4307 

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7420 

852i 

93i8 

9810 

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7 

54 

74i4 

99OO24 

2332 

4338 

6o4i 

744i 

8537|   9328 

9816 

9998 

6 

55 

746o 

oo65 

2368 

4369 

6067 

7462 

8552 

9339 

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9999 

5 

56 

75o6 

oio5 

2404 

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6093 

7482 

8568 

935o 

9827 

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4 

57 

755i 

oi46 

2439 

443o 

6118 

75o3 

8583 

936o 

9832 

I.OOOOO 

3  I 

58 

7597 

0187 

2475 

446  1 

6:44 

7523 

8599 

937o 

9837 

oooo 

2 

59 

7643 

0228 

25ll 

449i 

6169 

7544 

86i4 

938i 

9843 

oooo 

I 

9° 

8° 

7° 

6° 

5° 

4° 

3° 

2° 

1° 

0° 

1 

Natural  Co-sines. 

P>?,  0.80 

0.72 

o.63   o.55 

o.46   o.38 
' 

o.3o 

0.21 

o.i3 

o.o4 

' 

1 

NATURAL    TANGENTS. 


133 


1 

80° 

81° 

82°    83° 

84° 

85° 

86° 

87° 

88° 

89° 

o 

5.67i286.3i375 

7.n537i8.i4435 

9.5i436 

n.43oi 

i4.3oo7 

19.0811 

28.6363 

5-7.2900 

60 

I 

8094   2566j   3o42 

6398 

4io6 

4685 

3607 

1879 

877i 

58.2612 

69 

2 

9064!   3761 

4553 

837o 

679i 

5072 

4212 

2959 

29.1220 

59.2659 

58 

3 

6.70037!   4961 

6071 

8.20352 

9490 

546i 

4823 

4o5i 

37n 

6o.3o58 

67 

4 

ioi3:   6i65 

7594 

2344 

9.62205 

5853 

5438 

5i56 

6245 

61.3829 

56 

5 

1992'   7374 

9125 

43451"  4935 

6248 

6059 

6273 

8823 

62.4992 

55 

6 

2974   8587 

7.20661 

6355 

768ol   6645 

6685 

74o3 

3o.i446 

63.6567 

54 

7 

3960   9804 

22O4 

8376 

9.704411   7o45 

73i7 

8546 

4n6 

64.85&0 

53 

8 

4949^.41026 

3754 

8.3o4o6 

32  1  7;  7448 

7954   9702 

6833 

66.io55 

52 

9 

5g4i 

2253 

53io 

2446 

6009 

7853 

8596 

20.0872 

9599 

67.4019 

5i 

10 

~.7C937 

6.43484 

7.26873 

8.34496 

9.78817 

11.8262 

14.9244 

2O.2O56 

3i.24i6 

68.75oi 

5o 

1  1 

7936 

4720 

844s 

6555 

9.81641 

8673 

9898 

3253 

5284 

7o.i533 

49 

12 

8938 

596i 

7.3ooi8 

8625 

4482 

9087 

i5.o557 

4465 

82o5 

7i.6i5i 

48 

i3 

9944 

7206 

1600 

8.40705 

7338 

95o4 

1222 

569i 

32.n8i 

73.1390 

47 

i4 

5.8o953 

8456 

3i9o 

2795 

9.90211 

9923 

iSgS 

6932 

42  1  3 

74.7292 

46 

i5 

1966 

9710 

4786 

4896 

3ioi 

I2.o346 

2571 

8188 

73o3 

76.3900 

45 

16 

2982 

6.50970 

6389   7007 

6007 

0772 

3254 

946o 

33.o452 

78.1263 

44 

I? 

4ooi 

2234 

7999 

9128 

893i 

I2OI 

3943 

21.0747 

3662 

79.9434 

43 

i&-   6024 

35o3 

9616 

8.5i259 

10.0187 

i632 

4638 

2o49 

6935 

81.8470 

42 

19 

6o5i 

4777 

7.41240 

3402 

o483 

2067 

534o 

3369 

34.0273 

83.8435 

4i 

20 

5.87080 

6.56o55 

7.42871 

8.55555 

10.0780 

I2.25o5 

i5.6o48 

21.4704 

34.3678 

85.9398 

4o 

21 

8n4 

7339 

4509 

7718 

1080 

2946 

6762 

6o56 

7161 

88.i436 

39 

22 

9i5i 

8627 

6:54 

9893 

i38i 

3390 

7483 

7426 

35.0695 

90.4633 

38 

23 

5:9oi9i 

9921 

7806 

8.62078 

i683 

3838 

8211 

88i3 

43i3 

92.9085 

37 

24 

1236 

6.61219 

9465 

4275 

1988 

4288 

8945 

22.0217 

8006 

95.4895 

36 

25 

2283 

2523 

7.5n32 

6482 

2294 

4742 

9687 

i64o 

36.i776 

98.2179 

35 

26 

3335 

383i 

2806 

8701 

2602 

5i99 

i6.o435 

3o8i 

5627 

101.107 

34 

27 

439o 

5r44 

448  7 

8.70931 

2913 

566o 

II90 

454i 

956o 

104.171 

33 

28 

5448 

6463 

6176 

3l72 

3224 

6124 

1962 

6020 

37.3579 

107.426 

32 

29 

65io 

7787 

7872 

5425 

3538i   659i 

2722 

75l9 

•7686 

110.892 

3i 

3o 

5.97576 

6.691  16 

7.59575 

8.77689 

10.3854112.7062 

16.3499 

22.9o38 

38.1885 

114.689 

3o 

3i 

8646 

6.70450 

7.61287 

9964 

4172 

7536 

4283 

23.0577 

6i77 

n8.54o 

29 

32 

9720 

1789 

3oo5 

8.82252 

4491 

8oi4 

5o75 

2137 

39.o568 

122.774 

28 

33 

6.00797 

3i33 

4732 

455i 

48i3 

8496 

5874 

37i8 

5o59 

127.321 

27 

34 

1878 

4483 

6466 

6862 

5i36 

898i 

6681 

532i 

9655 

132.219 

26 

35 

2962 

5838 

8208 

9186 

5462 

9469 

7496 

6945 

4o.4358 

i37.5o7 

26 

36 

4o5i 

7199 

9957 

8.91520 

5789 

9962 

83i9 

8593 

9i74 

i43.237 

24 

3? 

5i43 

8564 

7.71715 

3867 

6118 

1  3.o458 

9i5o 

24.0263 

4i.4io6 

149-465 

23 

38 

6240 

9936 

348o 

6227 

645o 

o958 

999° 

i957 

9i58 

156.269 

22 

39 

734o 

6.8i3i2 

5254 

8598 

6783 

i46i 

I7.o837 

3675 

42.4335 

163.700 

21 

4o 

6.o8444 

6.82694 

7.77o35 

9.00983 

10.7119 

i3.i969 

I7.i693 

24.54i8 

42.9641 

171.886 

2O 

4i 

9552 

4082 

8825 

3379 

7457 

2480 

2558J   7185 

43.5o8i 

180.932 

J9 

42 

6.10664 

54?5 

7.  80622 

5789 

7797 

2996 

3432!   8978 

44.o66i 

190.984 

1  8 

43 

1779 

6874 

2428 

8211 

8139 

35i5 

43i4'25.o798 

6386 

202.219 

I7 

44 

2899 

8278 

424219.10646 

8483 

4o39 

6206.   2644 

45.2261 

2i4.858 

16 

45 

4o<>3 

9688 

6o64 

3o93 

8829 

4566 

6106 

45i7 

8294 

229.182 

i5 

46 

5i5i 

6.91104 

7895 

5554 

9178 

5o98 

7oi5 

64i8 

46.4489 

245.552 

i4 

4? 

628*3 

2525 

9734 

8028 

9529 

5634 

7934 

8348 

47.o853 

264.44i 

i3 

48 

7419 

3952 

7.9i582 

9.20616 

9882 

6i74 

8863|26.o3o7 

7395 

286.478 

12 

49 

8559 

5385 

3438 

3oi6 

1  1.0237 

6719 

9802 

2296 

48.4i2i 

312.521 

I  I 

5o 

6.19-703 

6.96823 

7.953o2 

9.2553o 

1  1.0594 

13.7267 

i8.o75o 

26.43l6 

49.1039 

343.774 

10 

5i 

6.zo85i 

8268 

7i76 

8o58 

0954 

7821 

1-708 

6367 

8i57 

381.971 

9 

52 

2003 

9718 

9o58 

9.30599 

i3i6 

8378 

2677 

845o 

5o.5485 

429.718 

8 

53 

3i6o 

7.on74 

8.oo948 

3i55 

1681 

894o 

3655 

27.0566 

5i.3o32 

491.106 

7 

54 

43si 

2637 

2848 

6724 

2048 

95o7 

4645 

2715 

52.o8o7 

672.967 

6  . 

55 

548G 

4io5 

4756I   83o7 

2417 

14.00-79 

5645 

4899 

8821 

687.549 

5 

56 

6655 

5579 

6674 

9.40904 

2789 

o655 

6656 

7117 

53.7o86 

859.436 

4 

5? 

7829 

7o59 

8600 

35i5|   3i63 

1235 

7678 

9372 

54.56i3 

1145.92 

3 

58 

9007 

8546 

8.io536 

6i4i:   354o'   1821 

87n 

28.1664 

55.44i5 

1718.87 

2 

59 

6  Hor«9 

7.ioo38l   2481 

8781   3919 

2411 

9?55|   3994 

56.35o6  3437.75 

I 

9° 

8° 

7° 

6° 

5° 

4° 

3° 

30   1   xo 

0° 

d 

Natural  Co-tangents. 

Jl 

& 

££17.80 

22.  10 

28.46 

37.83 

5.28 

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to  i  .  * 

! 

7 

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134 


JV  A  'T  u  R  A  L    SECANTS. 


Deg. 

0' 

10' 

20' 

30' 

40' 

50' 

P.  Fait 
tol'. 

o 

i  .000000 

i  .  000004 

i  .000017 

i  .oooo38 

i  .  000068 

i  .000106 

89 

2.5 

i 

000152 

000207 

000271 

ooo343 

000423 

ooo5i2 

88 

7.6 

a 

000609 

000715 

0008  3o 

000953 

001084 

001224 

87 

I2.7 

3 

001372 

001529 

001695 

001869 

OO2o5l 

002242 

86 

I7.8 

4 

002442 

oo265o 

002867 

003092 

oo3326 

003569 

85 

22.0 

5 

003820 

oo4o8o 

oo4348 

oo4625 

004911 

oo52o5 

84 

•6.1 

6 

oo55o8 

oo582o 

oo6i4i 

006470 

006808 

007154 

83 

33.3 

7 

007510 

007874 

008247 

008629 

009020 

009419 

82 

38.6 

8 

009828 

010245 

010671 

011106 

on55o 

OI2OO3 

81 

43.9 

9 

oi2465 

012936 

oi34i6 

013905 

oi44o3 

014910 

80 

49.3 

10 

1.015427 

1.015952 

1.016487 

i  .017030 

i.  017583 

I.  018145 

79 

54.8 

ii 

018717 

019297 

019887 

020487 

021095 

021713 

78 

60.4 

12 

022341 

022977 

023624 

024280 

024945 

O2562O 

77 

66.0 

i3 

O263o4 

026998 

027702 

O284i5 

029138 

029871 

76 

71,8 

i4 

o3o6i4 

o3i366 

032128 

032900 

033682 

034474 

75 

77-7 

i5 

035276 

o36o88 

036910 

037742 

o38584 

039437 

74 

83.7 

16 

040299 

041172 

o42o55 

042949 

o43853 

044767 

73 

89.8 

*7 

045692 

046627 

047573 

048529 

049496 

o5o4?4 

72 

96.1 

18 

o5i462 

o5246i 

o5347i 

054492 

o55524 

o56567 

71 

102.6 

19  '   057621 

o58686 

059762 

060849 

061947 

o63o57 

70 

109.2 

20 

1.064178 

i.o653io 

i.o68454 

i  .067609 

i  .068776 

1.069955 

69 

116.1 

21 

071145 

072347 

o7356i 

074786 

076024 

077273 

68 

123.2 

22 

078535 

079808 

081094 

082392 

083703 

o85o25 

67 

i3o.4 

23 

o8636o 

087708 

089068 

090441 

091827 

o93225 

66 

J37-9 

24 

o94636 

096060 

097498 

098948 

ioo4n 

101888 

65 

i45.6 

25     103378 

104881 

106398 

107929 

109473 

nioSo 

64 

I.S3.7 

26     II26O2 

114187 

115787 

i  i  7400 

119028 

120670 

63 

162.0 

27 

122326 

123997 

125682 

127382 

129096 

130826 

62 

170.7 

28 

132570 

i34329 

i36io4 

i37893 

139698 

i4i5i8 

61 

179.7 

29     143354 

i452o5 

147073 

148956 

i5o854 

152769 

60 

189.1 

3o 

1.154701 

i.  i  56648 

I.i586i2 

i  .160592 

1.162589 

i.i646o3 

59 

198.9 

3  1 

166633 

168681 

170746 

172828 

174927 

177044 

58 

209.  i 

32 

179178 

:8i33: 

i835oi 

i85689 

187895 

190120 

57 

219.7 

33 

192363 

194625 

196906 

199205 

2Ol523 

2o386i 

56 

230.9 

34 

206218 

208594 

210991 

2i34o6 

2i5842 

218298 

55 

242.6 

35 

220775 

223271 

225789 

228327 

23o886 

233466 

54 

254.8 

36  j   236o68 

238691 

24i336 

244oo3 

246691 

249402 

53 

267.7 

37    252i36 

254892 

257671 

260472 

263298 

266l46 

52 

281.3 

38    269018 

271914 

274834 

277779 

280748 

283741 

5i 

295.6 

39  »   286760 

289803 

292872 

295967 

2Q9o88 

302234 

5o 

310.7 

4o  i  i.3o54o7 

i.  308607 

i.3n833 

i.  3  1  5087 

i.3i8368 

I  .321677 

49 

326.7 

4i    325oi3 

328378 

331771 

335i92 

338643 

342123 

48 

343.6 

42 

345633 

349172 

352742 

356342 

359972 

363634 

47 

36i.5 

43 

367327 

371052 

374809 

378598 

382420 

386275 

46 

38o.5 

44   390164 

394086 

398042 

402032 

4o6o57 

4ion8 

45 

400.7 

45 

4i4ai4 

4i8345 

4225i3 

426718 

430960 

435239 

44 

422.3 

46 

439557 

443912 

4483o6 

45274O 

4572i3 

461726 

43 

445.3 

47 

466279 

470874 

475509 

480187 

484907 

489670 

42 

469.8 

48 

494477 

499327 

5o422I 

509160 

5i4i45 

519176 

4i 

496.2 

49 

524253 

529377 

534549 

539769 

545o38 

55o356 

4o 

524.4 

5o 

1.555724 

i  .56n42 

i  .5666i2 

I.572I34 

i.  5777o8 

1.583335 

39 

554.7 

5i 

589oi6 

594751 

6oo542 

6o6388 

612291 

6i825i 

38 

587.4 

52 

624269 

63o346 

636483 

642680 

648938 

655258 

37 

622.7 

53    66i64b 

668086 

674597 

68n73 

687815 

694524 

36 

660.9 

54  !   701  3o2 

708148 

7i5o64 

722O5I 

729110 

736241 

35 

702.2 

55    743447 

750727 

758o84 

7655i7 

773029 

780620 

34 

747-2 

56 

788292 

796045 

8o388i 

811801 

819806 

827899 

33 

796.2 

5? 

836078 

844348 

852707 

861159 

869704 

878344 

32 

849.8 

58 

887080 

895914 

904847 

9i388i 

923017 

982258 

3i 

908.5 

59 

941604 

95io58 

960621 

970294 

980081 

989982 

3o 

97?.  o 

i 

60' 

50' 

40' 

30' 

20' 

10' 

Deg. 

Natural  Co-secants. 

NATURAL   SECANT 


135 


f  Deg. 

0' 

10' 

20' 

30' 

40' 

50' 

P.  Port 
tol'. 

i  60 

2.OOOOOO 

2.o:oi36 

2,020393 

2.  03O772 

2.041276 

2.061906 

29 

io44 

6l 

062665 

o73556 

084679 

096739 

107036 

118474 

28 

1123 

62 

i3oo54 

141781 

i53655 

I6568I 

177869 

190196 

27 

1210 

63 

202680 

2i5346 

228168 

24ll58 

254320 

267667 

26 

i3o8 

64  '   281172 

294869 

308760 

322820 

337083 

35i542 

25 

1417 

65    366202 

38io65 

396i37 

4i  1421 

426922 

442645 

24 

i539 

66 

458593 

474773 

491187 

607843 

624744 

541896 

23 

1678 

67 

5593o5 

676976 

694914 

6i3i26 

63i6i8 

660396 

22 

i835 

68 

669467 

688387 

708614 

728604 

7488i4 

769453 

21 

2016 

69 

790428 

811747 

8334r9 

85545i 

877853 

90063.6 

20 

2222 

70 

2.923804 

2.947372 

2.971349 

2.996744 

3.020669 

3.o45835 

'9 

2461 

7* 

3.o7i553 

3.097736 

3.124396 

3.  i5i545 

179198 

207367 

18 

2740 

72 

236o68 

2653i5 

296123 

326610 

356490 

388082 

17 

3o68 

?3 

42o3o4 

453i73 

486711 

620937 

55587i 

59i536 

16 

3458 

74 

627955 

666162 

7o3i5r 

741978 

781660 

822226 

16 

3925 

75 

8637o3 

906126 

949622 

993929 

4.039380 

4.086913 

i4 

4492 

76 

4.133565 

4.182378 

4.232394 

4.283658 

3362i5 

390116 

i3 

6190 

77 

4454H 

602167 

56o4o8 

620226 

681676 

744821 

12 

6062 

78 

809734 

876491 

946169 

5.oi5852 

6.088628 

5.i63592 

II 

7171 

79 

5.24o843 

5.320486 

5.402633 

487404 

674926 

665333 

10 

8612 

80 

5.758770 

5.855392 

5.955362 

6.068868 

6.166067 

6.277193 

9 

81 

6.392453 

6.612081 

6.636329 

6.766469 

6.899794 

7.039622 

8 

82 

7.185297 

7.337191 

7.496711 

7.661298 

7.834433 

8.016646 

7 

83 

8.206609 

8.404669 

8.613790 

8.83367i 

9.o65i5i 

9.309170 

6 

84 

9.566772 

9.839123 

10.  12762 

10.43343 

10.76849 

11.10455 

5 

85 

11.47371 

11.86837 

12.29126 

12.74549 

i3.  23472 

i3.763n 

4 

86 

i4.33559 

14.96788 

16.63679 

i6.38o4r 

I7.i9843 

18.  10262 

3 

6; 

19.  10732 

2O.23O28 

21.49368 

22.92669 

24.66212 

26.46061 

2 

86 

28.65371 

3l.25758 

34.38232 

38.  20166 

42.97671 

49.11406 

I 

• 

89 

67.29869 

68.76736 

85.9456i 

114.6930 

i7i.8883 

343.7762 

O 

60' 

50' 

40' 

30' 

20' 

10' 

Deg. 

Natural  Co-secants. 

LENGTHS  OF  CIRCULAR  ARCS. 

Degrees. 

Minutes. 

Seconds. 

o 

I 

.oi74533 

0 

26 

.4537856 

o 

61 

.8901179 

/ 
I 

.0002909 

r 
I 

.  0000048 

2 

.0349066 

27 

.4712389 

62 

.9076712 

2 

.0006818 

2 

.0000097 

3 

.0623599 

28 

.4886922 

53 

.9260246 

3 

.0008727 

3 

.ooooi45 

4 

.0698132 

29 

.5o6i455 

54 

.9424778 

4 

.oon636 

4 

.0000194 

5 

.0872666 

3o 

.6236988 

55 

.9699311 

5 

.ooi4544 

5 

.0000242 

6 

.1047198 

ft  I 

.6410621 

56 

.  9773844 

6 

.0017453 

6 

.0000291 

7 

.1221730 

32 

.5585o54 

57 

•9948377 

7 

.0020362 

7 

.0000339 

8 

.i396263 

33 

.6769687 

58 

.0122910 

8 

.0023271 

8 

.oooo388 

9 

.1070796 

34 

.6934119 

59 

.0297443 

9 

.0026180 

9 

,oooo436 

10 

.1745329 

35 

.6108662 

60 

.0471976 

10 

.0029089 

10 

.0000486 

ii 

.  1919862 

36 

.6283i85 

65 

.  i  34464o 

ii 

.0031998 

ii 

.oooo533 

12 

.2094395 

37 

.6467718 

70 

.2217306 

12 

.0034907 

12 

.0000682 

:3 

.2268928 

38 

.6632261 

75 

.3089969 

i3 

.003/816 

i3 

.oooo63o 

i4 

.2443461 

39 

.6806784 

So 

.3962634 

i'4 

.0040724 

i4 

.0000679 

i5 

.2617994 

4o 

.6981317 

85 

.4836299 

16 

.oo43633 

i5 

.0000727 

16 

.2792627 

4i 

.  7166860 

90 

.6707963 

16 

.0046642 

16 

.0000776 

17 

.2967060 

42 

.733o383 

IOO 

.7453293 

*7 

.oo4g45i 

i? 

.0000824 

18 

.3i4r593 

43 

.7604916 

IIO 

.9198622 

18 

.0062360 

18 

.0000873 

'9 

.33i6i26 

44 

.7679449 

120 

2*>94395i 

J9 

.0066269 

!9 

.0000921 

20 

.3490669 

45 

.7863982 

i3o 

2.2689280 

20 

.0068178 

20 

.0000970 

21 

.3666191 

46 

.8028616 

i4o 

2.44346io 

25 

.0072722 

2.5 

.OOOI2I2 

22 

.3839724 

47 

.8203047 

i5o 

2.6179939 

3o 

.0087266 

36 

.0001454 

23 

.4014267 

48 

.837758o 

1  60 

2.7926268 

4o 

.0116355 

4o 

.0001939 

a/i 

.4188790 

49 

.8662113 

170 

2.9670.597 

5o 

.0145444 

5o 

.0002424 

25 

.4363323 

5o 

,8726646 

1  80 

3.  1416927 

60 

.01  745V- 

60   .  0002909 

136 


TRAVERSE    TABLE. 


Course 

Dist.  1. 

Dist.  2. 

Dist.  3. 

Dist.  4. 

Dist.  6. 

Lat. 

Dcp. 

Lat. 

Dep. 

Lat. 

Dep. 

1  Lat 

Dep. 

Lat 

Dep. 

o   / 

o    / 

o  i5 

1.  0000 

o.oo4 

2.OOOC 

0.008 

3.  OOOO 

o.oiS 

4-OOO 

0.017 

5.OOO 

O.02I 

89  45 

3o 

oooo 

008 

I.9999 

017 

2-9999 

0262 

3.999 

o34 

4-999 

o43 

3o 

4o 

0.9999 

oi3 

9998 

026 

9997 

0393 

999 

o52 

999 

o65 

:5 

I    0 

9998 

017 

9997 

o349 

9995 

o52^ 

999^1 

o698 

999 

o87 

89   o 

i5 

9998 

0218 

9995 

o436 

9993 

o654 

999c 

087 

9988 

IO9 

45 

3o 

9997 

0262 

9993 

o52; 

999° 

o785 

9986 

1047 

998 

i3o 

3o 

45 

9995 

o3o5 

9991 

061 

9986 

0916 

998 

1222 

9977 

i52 

i5 

2   O 

9994 

0349 

9988 

o698 

9982 

1047 

9976 

i396 

997° 

i74 

88   o 

.  i 

9992 

039: 

9985 

o785 

9977 

1178 

9969 

i57o 

996 

i96 

45 

3o 

999° 

o43  6 

9981 

0872 

997i 

iSog 

9962 

i745 

9952 

218 

3o 

45 

0.9988 

o.o48o 

1.9977 

0.0960 

2.9965 

o.i439 

3.9954 

o.i9i9 

4.994a 

o.2399 

i5 

3   o 

9986 

o523 

9973 

1047 

9959 

i57o 

9945 

209^ 

993i 

26l7 

87   o 

i5 

9984 

0567 

9968 

n34 

9952 

1701 

9936 

2268 

9920 

2835 

45 

3o 

9981 

0610 

9963 

1221 

9944 

i83i 

9925 

2442 

99°7 

3o52 

3o 

45 

9979 

o654 

9957 

i3o8 

9936 

1962 

99i4 

26l6 

9893 

327o 

i5 

4  o 

9976 

0698 

995i 

i395 

9927 

2093 

99o3 

2790 

9878 

3488 

86   o 

i5 

9973 

0741 

9945 

1482 

99i8 

2223 

989o 

296^ 

9863 

3705 

45 

3o 

9969 

o785 

9938 

i569 

99o8 

2354 

9877 

3i38 

9846 

3923 

3o 

45 

9966 

0828 

993i 

i656 

9897 

2484 

9863 

33i2 

9828 

4i4o 

i5 

5  o 

9962 

0872 

9924 

i743 

9886 

26i5 

9848 

3486 

9810 

4358 

85   o 

i5 

o.9958 

0.0915 

1.9916 

o.i83o 

2.9874 

0.2745 

3.9832 

o.366o 

4-979° 

o.4575 

45 

3o 

9954 

0958 

9908 

1917 

9862 

2875 

9816 

3834 

977° 

4792 

3o 

45 

995o 

1  002 

9899 

200^ 

9849 

3oo6 

9799 

4oo8 

9748 

5oo9 

i5 

6   o 

9945 

io45 

9890 

209I 

9836 

3i36 

9781 

4i8i 

9.726 

5226 

84  o 

T  *i 

9941 

1089 

9881 

2177 

9822 

3266 

9762 

4355 

97°3 

5443 

45 

3o 

9936 

ii3a 

9871 

226^ 

98o7 

3396 

9743 

4528 

9679 

566o 

3o 

45 

993i 

n75 

9861 

235i 

9792 

3526 

9723 

4701 

9653 

5877 

i5 

7   o 

9925 

1219 

985i 

2437 

9776 

3656 

9702 

4875 

9,627 

6o93 

83   o 

i5 

9920 

1262 

984o 

252^ 

976o 

3786 

9680 

5o48 

96oo 

63io 

45 

3o 

9914 

i3o5 

9829 

26ll 

9743 

39i6 

9658 

5221 

9572 

6526 

3o 

45 

0.9909 

o.i349 

1.9817 

o.2697 

.9726 

o.4o46 

.9635 

o.53o4 

.9543 

0.6743 

i5 

8   o 

99o3 

1392 

9805 

2783 

97o8 

4i75 

9611 

5567 

95i3 

6o59 

82   o 

i5 

9897 

1435 

9793 

2870 

969o 

43o5 

9586 

574o 

9483 

7175 

45 

3o 

9890 

1478 

9780 

2956 

967o 

4434 

956i 

59I2 

945i 

739o 

3o 

45 

9884 

l52I 

9767 

3o42 

965i 

4564 

9534 

6o85 

94i8 

7606 

i5 

9  ° 

9877 

1  564 

9754 

3l29 

963i 

4693 

95o8 

6257 

9384 

7822 

I    O 

i5 

9870 

1607 

9740 

3ai5 

96io 

4822 

948o 

643o 

q35o 

8o37 

45 

3o 

9863 

i65o 

9726 

33oi 

9589 

4g5i 

945i 

6602 

93i4 

8262 

3o 

45 

9856 

i693 

9711 

3387 

9567 

5o8o 

9422 

6774 

9278 

8467 

i5 

10   0 

9848 

I736 

9696 

3473 

9544 

5209 

9392 

6946 

9240 

8682 

o   o 

i5 

.9840 

0.1779 

.9681 

o.3559 

.952I 

0.5338 

.9362 

o.7n8 

.92O2 

.8897 

45 

3o 

9833 

1822 

9665 

3645 

9498 

5467 

933o 

7289 

9i63 

9112 

3o 

45 

9825 

i865 

9649 

373o 

94?4 

5596 

9298 

7461 

9I23 

9326 

i5 

II    0 

9816 

1908 

9633 

38i6 

9449 

5724 

9265 

7632 

9o8i 

954o 

9   ° 

i5 

9808 

1951 

9616 

3902 

94a4 

5853 

923l 

78o4 

9o39 

9755 

45 

3o 

9799 

i994 

9598 

3987 

9398 

598i 

9i97 

7975 

8996 

9968 

3o 

45 

979° 

2o36 

958i 

4o73 

937r 

6109 

9l62 

8i46 

8952 

.0182 

1  5 

12   0 

9781 

2079 

9563 

4i58 

9344 

6237 

9I26 

83x6 

89o7 

o396 

8   o 

i5 

9772 

2122 

9545 

4244 

93i- 

6365 

9o89 

8487 

8862 

0609 

45 

3o 

9763 

2l64 

9526 

4329 

9289 

6493 

9o52 

8658 

88i5 

0822 

3o 

45 

.9753 

.22O7 

.95o7 

.44  1  4 

.926o 

.6621 

.9oi4 

.8828 

.8767 

.io35 

i5 

i3   o 

9744 

225oj 

9487 

4499 

923l 

6749 

8975  8998 

8719 

1248 

7   o 

i5 

9734 

2292 

9468 

4584 

920I 

6876 

8935 

9i68 

8669 

i46o 

45 

3o 

9724 

2334 

9447 

4669 

9i7i 

7003 

8895 

9338 

8618 

1672 

3o 

45 

97i3 

2377 

9427 

4754 

9i4o 

7i3i 

8854 

95°7 

8567 

1884 

i5 

i4  o 

97o3 

2419 

94o6 

4838 

9I09 

7258 

8812 

9677 

85i5 

2096 

6   o 

i5 

9692 

2462 

9385 

4923 

9°77 

7385 

8769 

9846 

8462 

23o8 

45 

3o 

9681 

25o4 

9363 

5oo8 

9o44 

75n 

8726 

i.ooiS 

84o7 

2519 

3o 

45 

9670 

2546 

934i 

5o92 

9OI  I 

7638 

8682 

0184 

8352 

2730 

i5 

r5   o! 

9659 

2588 

93i9 

5i76 

8978 

7765 

8637 

o353 

8296 

2941 

5   o 

l!  Dep.    Lat. 

Dep.  |  Lat. 

Dep.  |  Lat. 

Dep. 

Lat. 

Dep. 

Lat 

Dist.  1. 

•i 

Dist.  2. 

ii 

Dist.  3.     Dist.  4. 

Dist.  5. 

Course. 

TRAVERSE    TABLE. 


137 


Course, 

Dist.  6. 

Dist.  7. 

Dist.  8. 

Dist.  9. 

Dist.  10. 

, 
i 

Lat. 

Dcp. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat 

Dep. 

o   / 

o    / 

o  i5 

6.9999 

0.0262 

6.9999 

o.o3o5 

7.9999 

o.o349 

8.9999 

o.o393 

9.9999 

o.o436 

89  45 

3o 

9998 

o524 

9997 

0611 

9997 

o698 

9997 

o785 

9996 

o873 

3o 

45 

9995 

0785 

9994 

o9i6 

9993 

1047 

9992 

1178 

999I 

i3o9 

i5 

I    0 

9991 

1047 

9989 

1222 

9988 

i396 

9986 

1571 

9985 

i745 

89   o 

i5 

9986 

1  309 

9983 

1527 

9981 

i745 

9979 

I963 

9976 

2181 

45 

3o 

9979 

1571 

9976 

i832 

9973 

2094 

9969 

2356 

9966 

2618 

3o 

45 

9972 

i832 

9967 

2i38 

9963 

2443 

9958 

2748 

9953 

3o54 

i5 

2    0 

9968 

2094 

9957 

2443 

995i 

2792 

9945 

3i4i 

9939 

349o 

83   o 

i5 

9954 

2356 

9946 

2748 

9938 

3i4i 

993i 

3533 

9923 

3926 

45 

3o 

9943 

2617 

9933 

3o53 

,9924 

349o 

99i4 

3926 

99°5 

4362 

3o 

45 

5.9931 

0.2879 

6.9919 

0.3358 

7.9908 

0.3838 

8.9896 

o.43i8 

9.9885 

o.4798 

i5 

3   o 

9918 

3i4o 

9904 

3664 

989o 

4i87 

9877 

4710 

9863 

5234 

87   o 

i5 

9904 

3402 

9887 

3968 

987i 

4535 

9855 

5lO2 

9839 

5669 

45 

3o 

9888 

3663 

9869 

4273 

985i 

4884 

9832 

5494 

98i3 

6io5 

3o 

45 

9872 

3924 

985o 

4578 

9829 

5232 

98o7 

5886 

9786 

654o 

i5 

4  o 

9854 

4i85 

9829 

4883 

98o5 

558i 

978i 

6278 

9756 

6976 

86   o 

i5 

9835 

4447 

9808 

5i88 

978o 

5929 

9753 

6670 

9725 

74n 

45 

3o 

9815 

4708 

9784 

5492 

9753 

6277 

9723 

7061 

9692 

7846 

3o 

45 

9794 

4968 

976o 

5797 

9725 

6625 

969i 

7453 

9657 

8281 

i5 

5   o 

9772 

5229 

9734 

6101 

9696 

6972 

9658 

7844 

9619 

8716 

85   o 

i5 

5.9748 

0.5490 

5.97o6 

o.64o5 

7.9664 

0.7320 

8.9622 

0.8235 

9.958o 

o.9i5o 

45 

3o 

9724 

575i 

9678 

6709 

9632 

7668 

9586 

8626 

954o 

9585 

3o 

45 

9698 

6011 

9648 

7013 

9597 

8oi5 

9547 

9oi7 

9497 

I.OOI9 

i5 

6   o 

9671 

6272 

96l7 

73i7 

9562 

8362 

95°7 

94o8 

9452 

o453 

84   o 

i5 

9643 

6532 

9584 

7621 

9525 

87o9 

9465 

9798 

94o6 

0887 

45 

3o 

9614 

6792 

955o 

7924 

9486 

9o56 

942I 

1.0188 

9357 

1320 

3o 

45 

9584 

7052 

95i5 

8228 

9445 

94o3 

9376 

o578 

93o7 

1754 

i5 

7   o 

9553 

7312 

9478 

853i 

94o4 

975o 

9329 

o968 

9255 

2187 

83   o 

i5 

9520 

7672 

944o 

8834 

936o 

i.oo96 

928o 

i358 

920O 

2620 

45 

3o 

9487 

7832 

94oi 

9i37 

93i6 

o442 

923o 

i747 

9i44 

3o53 

3o 

45 

5.9452 

0.8091 

6.936i 

o.944o 

7.9269 

1.0788 

8.9i78 

I.2l37 

9.9o87 

1.3485 

i5 

8  o 

9416 

835o 

93r9 

9742 

922I 

n34 

9I24 

2526 

9027 

39i7 

82   o 

i5 

9379 

8610 

9276 

i.oo44 

9I72 

1479 

9o69 

29l4 

8965 

4349 

45 

3o 

934i 

8869 

923l 

o34? 

9I2I 

1825 

9on 

33o3 

8902 

4781 

3o 

45 

93o2 

9127 

9i85 

o649 

9o69 

2170 

8953 

369i 

8836 

5212 

i5 

9  ° 

926l 

9386 

9i38 

oa5o 

9oi5 

25i5 

8892 

4o79 

8769 

5643 

81   o 

i5 

922O 

9645 

9o9o 

1252 

896o 

2859 

883o 

4467 

87oo 

6074 

45 

3o 

9i77 

99o3 

9o4o 

i553 

89o3 

3204 

8766 

4854 

8629 

65o5 

3o 

45 

9i33 

1.0161 

8989 

i854 

8844 

3548 

8700 

524i 

8556 

6935 

i5 

IO    O 

9o88 

0419 

8937 

2i55 

8785 

3892 

8633 

5628 

.848  1 

7365 

80   o 

i5 

5.9o42 

1.0677 

6.8883 

1.2456 

7.8723 

1.4235 

8.8564 

i.6oi5 

9.84o4 

i.7794 

45 

3o 

8995 

0934 

8828 

2756 

8660 

4579 

8493 

64oi 

8325 

8224 

3o 

45 

8947 

1191 

8772 

3o57 

8596 

4922 

8421 

6787 

8245 

8652 

i5 

II    0 

8898 

1  449 

8714 

3357 

853o 

5265 

8346 

7i73 

8  1  63 

9o8i 

79   o 

i5 

8847 

1705 

8655 

3656 

8463 

5607 

8271 

7558 

8o79 

95o9 

45 

3o 

8795 

1962 

8595 

3956 

8394 

5949 

8i93 

7943 

7992 

9937 

3o 

45 

8743 

2219 

8533 

4255 

8324 

6291 

8ii4 

8328 

79°5 

2.o364 

i5 

12    O 

8689 

2475 

8470 

4554 

8252 

6633 

8o33 

8712 

78i5 

O79i 

78   o 

i5 

8634 

2731 

84o6 

4852 

8178 

6974 

795i 

9o96 

7723 

1218 

45 

3o 

8578 

2986 

834i 

5i5i 

8104 

73i5 

7867 

948o 

763o 

1  644 

3o 

45 

5.8521 

1.3242 

6.8274 

r.5449 

7.8027 

1.7656 

8.7781 

i.9863 

9.7534 

2.2070 

i5 

i3   o 

8462 

3497 

8206 

5747 

795o 

7996 

7693 

2.0246 

7437 

2495 

77   ° 

i5 

84o3 

3752 

8i37 

6o44 

7870 

8336 

76o4 

0628 

7338 

2920 

45 

3o 

8342 

4007 

8066 

634i 

779° 

8676 

75i3 

1010 

7237 

3345 

3o 

45 

8281 

4261 

7994 

6638 

7707 

9015 

742I 

i392 

7i34 

3769 

i5 

i4  o 

8218 

45i5 

792I 

6935 

7624 

9354 

7327 

1773 

7D3o 

4l92 

76   o 

i5 

8i54 

4769 

7846 

723l 

7538 

9692 

723l 

2  I  54 

6923 

46i5 

45 

3o 

8089 

5o23 

7770 

7527 

7452 

2.oo3o 

7i33 

2534 

68i5 

5o38 

5o 

45 

8023 

5276 

7693 

-7822 

7364 

o368 

7o34 

29I^ 

67o5 

546o 

i5 

i5   o 

7956 

5529 

7616 

8117 

7274 

0706 

6933 

3294 

6593 

5882 

75   o 

Dep. 

Lat. 

Dep. 

Lat. 

Dep.  |  Lat. 

Dep. 

Lat."1 

Dep. 

Lat 

Dist.  6. 

Dist.  7. 

Dist.  8. 

Dist.  9. 

Dist  10. 

Coursa 

138 


TRAVERSE    TABLE. 


Course. 

Dist.  1. 

Dist.  2. 

Dist.  3. 

Dist.  4. 

Dist  5. 

Lat. 

Dep. 

Lat 

Dep. 

Lat. 

Dep. 

Lat 

Dcp. 

f  Lat 

Dep. 

0     f 

o    / 

i5  i5 

0-9648 

0.2630 

1.9296 

0.5261 

2.8944 

o.789i 

3.859i 

I.052I 

4.8239 

I.3i52 

74  45 

3o 

9636 

2672 

9273 

5345 

89o9 

8oi7 

8545 

o69o 

8182 

3362 

3o 

45 

9625 

2714 

9249 

5429 

887; 

8i43 

8498 

o858 

8i23 

3572 

i5 

16  o 

q6i3 

2756 

9225 

55i3 

8838 

8269 

845o 

IO25 

8o63 

3782 

74   o 

i5 

9600 

2798 

9201 

5597 

8801 

8395 

8402 

n93 

8002 

399i 

45 

3o 

9588 

2840 

9176 

568o 

8765 

8520 

8353 

i36i 

794i 

4201 

So 

45 

9576 

2882 

9i5i 

5764 

8727 

8646 

83o3 

i528 

7879 

44io 

i5 

17   o 

9563 

292^ 

9126 

5847 

8689 

877i 

8262 

i695 

78i5 

46i9 

73   o 

i5 

955o 

2966 

9100 

5931 

865i 

8896 

8201 

1862 

775i 

4827 

45 

3o 

9537 

3007 

9074 

6oi/ 

8612 

9O2I 

8i49 

2028 

7686 

5o35 

3o 

45 

0.9524 

o.3o49 

1.9048 

0.6097 

2.8572 

o.9i46 

3.8o96 

I.2I95 

4.7620 

1.5243 

i5 

18   o 

95n 

3090 

9021 

6180 

8532 

927I 

8042 

236i 

7553 

545  1 

72   o 

i5 

9497 

3i32 

8994 

6263 

849i 

9395 

7988 

2527 

7485 

5658 

45 

3o 

9483 

3i73 

8966 

6346 

845o 

95i9 

7933 

2692 

74i6 

5865 

3o 

45 

9469 

32i4 

8939 

6429 

84o8 

9643 

7877 

2858 

7347 

6072 

i5 

19   o 

9455 

3^56 

8910 

65n 

8366 

9767 

7821 

3o23 

7276 

6278 

71   o 

i5 

944  1 

3297 

8882 

6594 

8323 

989i 

7?64 

3i88 

7204 

6485 

45 

3o 

9426 

3338 

8853 

6676 

8279 

i.ooi4 

7706 

3352 

7l32 

669o 

3o 

45 

9412 

3379 

8824 

6758 

8235 

oi38 

7647 

35i7 

7o59 

6896 

i5 

20   o 

9397 

3420 

8794 

684o 

8i9i 

0261 

7588 

368i 

6985 

7101 

70   o 

i5 

0.9382 

0.346: 

1.8764 

0.6922 

2.8146 

i.o384 

3.7528 

1.3845 

4.69io 

1.7306 

45 

3o 

9367 

35o2 

8733 

7004 

8100 

o5o6 

7467 

4oo8 

6834 

75io 

3o 

45 

935i 

3543 

8703 

7086 

8o54 

o629 

74o5 

4172 

6757 

77i5 

i5 

21    0 

9336 

3584 

8672 

7167 

8007 

0751 

7343 

4335 

6679 

79i8 

69   o 

i5 

9320 

3624 

864o 

7249 

796o 

0873 

7280 

4498 

6600 

8122 

45 

3o 

93o4 

3665 

8608 

733o 

79i3 

o995 

7217 

466o 

652i 

8325 

3o 

45 

9288 

37o6 

8576 

7411 

7864 

1117 

7l52 

4822 

644o 

8528 

i5 

22    O 

9272 

3y46 

8544 

7492 

7816 

1238 

7087 

4984 

6359 

873o 

68   o 

i5 

9255 

3786 

85n 

7573 

7766 

i359 

7022 

5i46 

6277 

8932 

45 

3o 

9239 

3827 

8478 

7654 

7716 

i48i 

6955 

53o7 

6i94 

9i34 

3o 

45 

0.9222 

0.3867 

1.8444 

o.7734 

2.7666 

1.1601 

3.6888 

1.5468 

4.6110 

1.9336 

i5 

23    0 

9205 

39o7 

84io 

78i5 

76i5 

1722 

6820 

5629 

6025 

9537 

67   o 

l5 

9188 

3947 

8376 

7895 

7564 

1842 

6752 

579o 

594o 

9737 

45 

3o 

9171 

3987 

834i 

7975 

7512 

i962 

6682 

595o 

5853 

9937 

3o 

45 

9i53 

4027 

83o6 

8o55 

7459 

2082 

6612 

6110 

5766 

2.0l37 

i5 

24    0 

9i35 

4o67 

8271 

8i35 

74o6 

2202 

6542 

6269 

5677 

o337 

66   o 

i5 

9118 

4107 

8235 

8214 

7353 

2322 

6470 

64aq 

5588 

o536 

45 

3o 

9100 

4i47 

8i99 

8294 

7299 

244i 

6398 

6588 

5498 

o735 

3o 

45 

9081 

4187 

8i63 

8373 

7244 

2660 

6326 

6746 

54o7 

o933 

i5 

26   o 

9063 

4226 

8126 

8452 

7i89 

2679 

6252 

69o5 

53i5 

u3i 

65   o 

i5 

0.9045 

0.4266 

i.8o89 

o.853i 

2.7i34 

i.2797 

3.6178 

1.7063 

4.5223 

2.1328 

45 

3o 

9026 

43o5 

8o52 

8610 

7078 

29l5 

6io3 

•7220 

5i29 

i526 

3o 

45 

9007 

4344 

8oi4 

8689 

7021 

3o33 

6028 

7378 

5o35 

I722 

i5 

26   o 

8988 

4384 

7976 

8767 

6964 

3i5i 

5952 

7535 

494o 

1919 

64   o 

i5 

8969 

4423 

7937 

8846 

69o6 

3269 

5875 

•7692 

4844 

2Il4 

45 

3o 

8949 

4462 

7899 

8924 

6848 

3386 

5797 

7848 

4747 

23lO 

3o 

45 

8930 

45oi 

7860 

9002 

6789 

35o3 

57i9 

8oo4 

4649 

25o5 

i5 

27   0 

8910 

454o 

7820 

9080 

673o 

3620 

564o 

8160 

455o 

2700 

63   o 

i5 

8890 

4579 

7780 

9l57 

667i 

3736 

556i 

83:5 

445  1 

2894 

45 

3o 

8870 

46i7 

7?4o 

9235 

6610 

3852 

548o 

847o 

43  5  1 

3o87 

3o 

45 

o.885o 

0.4656 

1.7700 

0.9312 

2.655o 

1.3968 

3.54oo 

1.8625 

44249 

2.3281 

i5 

28   o 

8829 

4695 

7659 

9389 

6488 

4o84 

53i8 

8779 

4i47 

3474 

62   o 

16 

8809 

4733 

7618 

9466 

6427 

4200 

5236 

8933 

4o45 

3666 

45 

3o 

8788 

4772 

7576 

9543 

6365 

43i5 

5i53 

9o86 

394i 

3858 

3o 

45 

8767 

48io 

7535 

9620 

63o2 

443o 

5o69 

924o 

3836 

4049 

i5 

29   o 

8746 

4848 

7492 

9696 

6239 

4544 

4985 

9392 

373i 

4240 

61   o 

i5 

8725 

4886 

745o 

9772 

6i75 

4659 

49oo 

9545 

3625 

443  1 

45 

3o 

8704 

4924 

7407 

9848 

61  1  1 

4773 

48i4 

9697i 

35i8 

4621 

3o 

45 

8682 

4962 

7364 

9.024 

6o46 

4886 

4728 

9849| 

34io 

48n 

i5 

3o   o 

8660 

5ooo 

7321 

I.OOOO 

598i 

5ooo 

464  1 

2.000O 

33oi 

5ooo 

60   o 

Dop. 

Lat 

Dcp. 

Lat 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat 

Dist.  1. 

Dist.  2. 

Dist.  3. 

Dist.  4. 

Dist.  5. 

Course. 

TRAVERSE    TABLE. 


139 


Course. 

Dist.  6. 

Dist.  7. 

Dist.  8. 

Dist.  9. 

Dist.  10. 

. 

Lat. 

Dep. 

Lat 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat 

Dep. 

c  r 

o    / 

i5  i5 

5.7887 

1.5782 

6.7535 

i.84i2 

7.7i83 

2.IO42 

8.683i 

2.3673 

9-6479 

2.63o3 

74  45 

3o 

78l8 

6o34 

7454 

87o7 

7o9o 

i379 

6727 

4o5i 

6363 

6724 

3o 

45 

7747 

6286 

7372 

9001 

6996 

I7i5 

6621 

443o 

6246 

7l44 

1  5 

16   o 

7676 

6538 

•7288 

9295 

69oi 

2o5i 

65i4 

48o7 

6126 

7564 

74  o 

i5 

76o3 

6790 

72O3 

9588 

68o4 

2386 

64o4 

5i85 

6oo5 

7983 

45 

3o 

7529 

704  1 

7117 

9881 

67o6 

272I 

6294 

556i 

5882 

8402 

3o 

45 

7454 

7292 

7o3o 

2.0I74 

6606 

3o56 

6181 

5938 

5757 

8820 

i5 

17   o 

7378 

7542 

694i 

o466 

65o4 

339o 

6067 

63i3 

563o 

9237 

73   o 

i5 

73oi 

7792 

685i 

o758 

6402 

3723 

5952 

6689 

55o2 

9654 

45 

3o 

7223 

8042 

6760 

1049 

6297 

4o56 

5835 

7o64 

5372 

3.oo7i 

3o 

45 

5.7i44 

1.8292 

6.6668 

2.i34i 

7.6i92 

2.4389 

8.57i6 

2.7438 

9.5240 

3.o486 

i5 

18   o 

7063 

854i 

6574 

i63i 

6o85 

472I 

5595 

78l2 

5io6 

O902 

72   o 

i5 

6982 

8790 

6479 

1921 

5976 

5o53 

5473 

8i85 

4g7o 

i3i6 

45 

3o 

6899 

9o38 

6383 

2211 

5866 

5384 

5349 

8557 

4832 

I73o 

3o 

45 

6816 

9286 

6285 

25oi 

5754 

57i5 

5224 

893o 

4693 

2i44 

i5 

19   o 

673i 

9534 

6186 

279° 

564i 

6o45 

5o97 

93oi 

4552 

2557 

7I    0 

i5 

6645 

9781 

6086 

3o78 

5527 

6375 

4968 

9672 

4409 

2969 

45 

3o 

6558 

2.0028 

5985 

3366 

54n 

67o5 

4838 

3.0043 

4264 

338i 

3o 

45 

6471 

0275 

5882 

3654 

5294 

7o33 

4706 

o4i3 

4n8 

3792 

i5 

20   o 

6382 

o52I 

5778 

394i 

5i75 

7362 

4572 

0782 

3969 

4202 

70   o 

i5 

5.6291 

2.0767 

6.5673 

2.4228 

7.5o55 

2.7689 

8.4437 

3.ii5i 

9.38i9 

3.46i2 

45 

3o 

6200 

1012 

5567 

45i5 

4934 

8oi7 

43oo 

i5i9 

3667 

5021 

3o 

45 

6108 

1257 

5459 

48oo 

48n 

8343 

4162 

1886 

35i4 

5429 

i5 

21    O 

6016 

i5o2 

535i 

5o86 

4686 

8669 

4022 

2253 

3358 

5837 

6g   o 

i5 

5920 

1746 

524i 

537i 

456i 

8995 

388i 

26i9 

3201 

6244 

45 

3o 

5825 

1990 

5129 

5655 

4433 

932O 

3738 

2986 

3o4a 

665o 

3o 

45 

5729 

2233 

5oi7 

5939 

43o5 

9645 

3593 

335o 

2881 

7o56 

i5 

22    O 

563i 

2476 

4go3 

6222 

4i75 

9969 

3447 

37i5 

2-718 

746i 

68   o 

i5 

5532 

2719 

4788 

65o5 

4o43 

3.O292 

3299 

4o78 

2554 

7865 

45 

3o 

5433 

2961 

4672 

6788 

39io 

o6i5 

3i49 

4442 

2388 

8268 

3o 

45 

5.5332 

2.3203 

6.4554 

2.7070 

7-3776 

3.o937 

8.2998 

3.48o4 

9.2220 

3.867i 

i5 

23   0 

523o 

3444 

4435 

735i 

364o 

1258 

2845 

5i66 

2o5o 

9o73 

67   o 

i5 

6127 

3585 

43i5 

7632 

35o3 

i58o 

2691 

5527 

i879 

9474 

45 

3o 

5o24 

3925 

4194 

79I2 

3365 

1900 

2535 

5887 

I7o6 

9875 

3o 

45 

4919 

4i65 

4o72 

8i92 

3225 

2220 

2378 

6247 

i53i 

4.0275 

i5 

24    0 

48i3 

44o4 

3g48 

8472 

3o84 

2539 

22I9 

6606 

i355 

o674 

66   o 

i5 

4706 

4643 

3823 

875o 

294i 

2858 

2o59 

6965 

n76 

IO72 

45 

3o 

4598 

4882 

3697 

9O29 

2797 

3i75 

i897 

7322 

o996 

i469 

3o 

45 

4489 

5l2O 

357o 

93o6 

265i 

3493 

I733 

7679 

0814 

1866 

i5 

25   0 

4378 

5357 

3442 

9583 

25o5 

38o9 

i568 

8o36 

o63i 

2262 

65   o 

i5 

5.4267 

2.5594 

6.33i2 

2.986o 

7.2356 

3.4i25 

8.i4oi 

3.8391 

9.o446 

4.2657 

45 

3o 

4i55 

583i 

3i8i 

3.oi36 

2207 

444  1 

1233 

8746 

os59 

3o5i 

3o 

45 

4042 

6067 

3o49 

o4n 

2o56 

4756 

io63 

9100 

oo7o 

3445 

i5 

26   o 

3928 

63o2 

2916 

0686 

1904 

5o7o 

o89i 

9453 

8.9879 

3837 

64   o 

i5 

38i2 

6537 

278l 

o96o 

I75o 

5383 

o7i9 

9806 

9687 

4229 

45 

3o 

3696 

6772 

2645 

1234 

i595 

5696 

o544 

4-oi58 

9493 

4620 

3o 

45 

3579 

7006 

2509 

i5o7 

i438 

6008 

o368 

oSog 

9298 

5oio 

i5 

27   o 

346o 

7239 

237O 

I779 

1281 

63i9 

0191 

0859 

9IOI 

5399 

63   o 

i5 

334i 

7472 

223l 

2o5i 

II2I 

663o 

0012 

1209 

89O2 

5787 

45 

3o 

3221 

77°5 

2091 

2322 

0961 

6g4o 

7.983i 

i557 

87oi 

6i75 

3o 

45 

D.Sogg 

2-7937 

6.1949 

3.2593 

7-°799 

3.7249 

7.9649 

4.1905 

8.8499 

4.656i 

i5 

28   o 

2977 

8168 

1806 

2863 

o636 

7558 

9465 

2252 

8295 

6947 

62   o 

i5 

2853 

8399 

l662 

3i32 

o47i 

7866 

9280 

2599 

8089 

7332 

45 

3o 

2729 

863o 

i5i7 

34oi 

o3o5 

8i73 

9094 

2944 

•7882 

77i6 

3o 

45 

2604 

8859 

i37i 

3669 

oi38 

8479 

SgoS 

3289 

7673 

8o99 

i5 

20    0 

2477 

9089 

1223 

3937 

6.997o 

8785 

87i6 

3633 

7462 

848i 

61   o 

i5 

235o 

93i7 

io75 

4203 

9800 

9090 

8525 

3976 

725o 

8862 

45 

3o 

2221 

9545 

0925 

447o 

9628 

9394 

8332 

43i8 

7o36 

9242 

3o 

45 

2092 

9773 

o774 

4735 

9456 

969-7 

8i38 

4659 

6820 

9622 

i5 

3o  o 

1962 

3.OCOO 

0622 

5ooo 

9282 

4.0000 

7942 

5ooo 

66o3 

S.oooo 

60   o 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dist.  6. 

Dist.  7. 

Dist.  8. 

Dist.  9. 

Dist.  10. 

Oourse. 

1  40 


TRAVERSE    TABLE. 


Bourse. 

Dist.  1. 

Dist.  2. 

Dist.  3. 

Dist.  4. 

Dist.  5.  1 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat 

Dep. 

Lat 

Dep. 

o    / 

o    / 

So  i5 

0.8638 

o.5o38 

1.7277 

1.0076 

2.59l5 

I.5ll3 

3.4553 

2.Ol5l 

4.3192 

2.5l89 

59  45 

3o 

8616 

5o75 

7233 

Ol5l 

5849 

5226 

4465 

0302 

3o8i 

5377 

3o 

45 

8594 

5n3 

7188 

O226 

5782 

5339 

4376 

o452 

2970 

5565 

i5 

h   o 

8572 

5i5o 

7i43 

o3oi 

57I5 

545i 

4287 

0602 

2858 

5752 

59   o 

i5 

8549 

5:88 

7098 

o375 

5647 

5563 

4i96 

o75i 

2746 

5939 

45 

3o 

8520 

5225 

7o53 

o45o 

5579 

5675 

4/o6 

o9oo 

2632 

6125 

3o 

45 

85o4 

5262 

7007 

o524 

55n 

5786 

4oi4 

io49 

25i8 

63n 

i5 

32    0 

848o 

5299 

6961 

o598 

544i 

5898 

3922 

II97 

2402 

6496 

58   o 

i5 

8457 

5336 

6915 

0672 

5372 

6008 

3829 

i345 

2286 

6681 

45 

3o 

8434 

5373 

6868 

0746 

53o2 

6119 

3736 

l492 

2170 

6865 

3o 

45 

o.84io 

o.54io 

1.6821 

1.0819 

2.5231 

1.6229 

3.3642 

2.1639 

4«2052 

2.  7049 

i5 

^3   o 

8387 

5446 

6773 

0893 

5i6o 

6339 

3547 

i786 

1934 

7232 

57   o 

i5 

8363 

5483 

6726 

0966 

5o89 

6449 

345i 

1982 

1814 

74i5 

45 

3o 

8339 

55i9 

6678 

1039 

5oi7 

6558 

3355 

2077 

1694 

7597 

3o 

45 

83i5 

5556 

6629 

ii  1  1 

4g44 

6667 

3259 

2223 

1573 

7779 

ID 

34  o 

8290 

5592 

658i 

u84 

487i 

6776 

3i62 

2368 

i452 

7960 

56   o 

i5 

8266 

5628 

6532 

1256 

4798 

6884 

3o64 

25l2 

1329 

8i4o 

45 

3o 

8241 

5664 

6483 

i328 

4-724 

6992 

2966 

2656 

1206 

8320 

3o 

45 

8216 

57oo 

6433 

i4oo 

4649 

7100 

2866 

2800 

1082 

85oo 

i5 

35   o 

8192 

5736 

6383 

1472 

4575 

7207 

2766 

2943 

0958 

8679 

55   o 

i5 

0.8166 

o.577i 

1.6333 

i.i543 

2.4499 

i.73i4 

3.2666 

2.3o86 

4.o832 

2.8857 

45 

3o 

8i4i 

58o7 

6282 

1614 

4423 

7421 

2565 

3228 

0706 

9o35 

3o 

45 

8116 

5842 

623i 

i685 

4347 

7527 

2463 

337o 

°579 

92I2 

i5 

36  o 

8090 

5878 

6180 

i756 

427I 

7634 

236i 

35n 

o45i 

9389 

54   o 

i5 

8064 

59i3 

6129 

1826 

4193 

7739 

2258 

3652  j  o322 

9565 

45 

3o 

8o39 

5948 

6077 

1896 

4n6 

7845 

2i54 

3793 

oi93 

9741 

3o 

45 

8oi3 

5983 

6026 

1966 

4o38 

795o 

2o5o 

3933 

oo63 

9916 

ID 

87  o 

7986 

6018 

5973 

2o36 

3959 

8o54 

1945 

4o73 

3.9932 

3.oo9i 

53   o 

i5 

7960 

6o53 

6920 

2106 

388o 

8i59 

i84o 

4212 

9800 

0265 

45 

3o 

7934 

6088 

5867 

2175 

38oi 

8263 

I734 

435o 

9668 

o438 

3o 

45 

0.7907 

0.6122 

i.58r4 

1.2244 

2.372I 

1.8867 

3.1628 

2.4489 

3.9534 

3.o6i  i 

i5 

33   o 

7880 

6i57 

576o 

23i3 

364o 

8470 

l52O 

4626 

9401 

0783 

52    0  : 

i5 

7853 

6191 

57o6 

2382 

356o 

8573 

i4i3 

4764 

9266 

o955 

45 

3o 

7826 

6225 

5652 

245o 

3478 

8675 

i3o4 

49oi 

9i3o 

1126 

3o 

45 

7799 

6259 

5598 

2618 

3397 

8778 

1  195 

5o37 

8994 

1296 

i5 

3o   o 

7771 

6293 

5543 

2586 

33i4 

8880 

1086 

5i73 

8857 

i466 

5i   o 

i5 

7744 

6327 

5488 

2654 

3232 

8981 

0976 

53o8 

8720 

i635 

45 

3o 

7716 

636i 

5432 

2722 

3i49 

9082 

o865 

5443 

858i 

1804 

3o 

45 

7688 

6394 

5377 

2789 

3o65 

9i83 

o754 

5578 

8442 

1972 

i5 

4o   o 

7660 

6428 

532i 

2856 

2981 

9284 

0642 

57I2 

8302 

2139 

5o   o 

i5 

0.7632 

o.646i 

i.5265 

1.2922 

2.2897 

1.9884 

3.o529 

2.5845 

3.8162 

3.23o6 

45 

3o 

7604 

64g4 

5208 

2989 

2812 

9483 

o4i6 

5978 

8020 

2472 

3o 

45 

7576 

6528 

5i5i 

3o55 

2727 

9583 

o3o3 

61  10 

7878 

2638 

i5 

4i  o 

7547 

656r 

5094 

3l2I 

264l 

9682 

0188 

6242 

7735 

28o3 

49   o 

i5 

75i8 

6593 

5o37 

3i87 

2555 

9780 

oo74 

6374 

7592 

2967 

45 

3o 

7490 

6626 

4979 

3252 

2469 

9879 

2.9958 

65o5 

7448 

3i3i 

3o 

45 

746  1 

6659 

4921 

33i8 

2382 

9976 

9842 

6635 

73o3 

8294 

i5 

42    0 

743  1 

6691 

4863 

3383 

2294 

2.0074 

9726 

6765 

7i57 

3457 

48   o 

i5 

7402 

6724 

48o4 

3447 

22O7 

01  7  I 

9609 

6895 

•7011 

36i8 

45 

3o 

7373 

6756 

4746 

35i2 

2118 

0268 

9491 

7024 

6864 

378o 

3o 

45 

o.7343 

o.6788 

1.4686 

i.3576 

2.2O30 

2.o364 

2.9373 

2.7152 

3.67i6 

3.394o 

i5 

43  o 

73i4 

6820 

4627 

364o 

I94l 

o46o 

9254 

7280 

6568 

4ioo 

47  o 

i5 

7284 

6852 

4567 

37o4 

i85i 

o555 

9i35 

7407 

6419 

4259 

45 

3o 

7254 

6884 

45o7 

3767 

1761 

o65i 

9oi5 

7534 

6269 

44i8 

3o 

45 

7224 

69i5 

444  7 

383o 

1671 

o745 

8895 

7661 

6118 

4576 

i5 

44  o 

7i93 

6947 

4387 

3893 

i58o 

o84o 

8774 

7786 

5967 

4733 

46   o 

i5 

7i63 

6978 

4326 

3956 

1489 

o934 

8652 

7912 

58i5 

489o 

45 

3o 

7i33 

7°°9 

4265 

4oi8 

i3g8 

I027 

853o 

8o36 

5663 

5o45 

3o 

45 

7102 

7o4o 

4204 

4o8o 

i3o6 

1120 

84o7 

8161 

55o9 

6201 

i5 

45  o 

7o7i 

7071 

4i42 

4i42 

I2l3 

I2l3 

8284 

8284 

5355 

5355 

45   o 

Dep. 

Lat 

Dep.    Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat 

Dist.  1. 

Dist.  2. 

Dist.  3. 

Dist  4. 

Dist.  5. 

Course. 

TRAVERSE    TABLE. 


141 


Course. 

Dist.  6. 

Dist.  7.  ||  Dist.  8. 

Dist.  9. 

Dist.  10. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

o   / 

o  r 

30  ID 

5.i83o 

3.0226 

6.o468 

3.5264 

6.9IO7 

4.0302 

7.7745 

4.5340 

8.6384 

5.o377 

59  45 

3o 

1698 

0452 

o3i^ 

5528 

893o 

o6o3 

7547 

5678 

6i63 

o754 

3o 

45 

1  564 

,  0678 

oi58 

579i 

8753 

o9o3 

7347 

6016 

594i 

1129 

i5 

3i   o 

i43o 

0902 

OOO2 

6o53 

8573 

1203 

7145 

6353 

57i7 

i5o^ 

59   o| 

i5 

I295 

1126 

5.9844 

63i4 

8393 

1502 

6942 

669o 

549i 

1877 

45 

3o 

n58 

i35o 

9685 

6575 

8211 

1800 

6738 

7025 

5264 

2250 

3o 

45 

IO2I 

i573 

9525 

6835 

8028 

ao97 

6532 

7359 

5o35 

2621 

i5 

32   0 

o883 

i795 

9363 

7094 

7844 

2394 

6324 

7693 

48o5 

2992 

58   o 

1  5 

0744 

2017 

92OI 

7353 

7658 

2689 

6116 

8o25 

4573 

336i 

45 

3o 

o6o3 

2238 

9°37 

7611 

7471 

2984 

59o5 

8357 

4339 

373o 

3o 

45 

5.o462 

3.2458 

5.8873 

3.7868 

6.7283 

4.3278 

7.5694 

4.8688 

8.4io4 

5.4097 

i5 

33   o 

O32O 

2678 

8707 

8i25 

7094 

357i 

548o 

9oi8 

3867 

4464 

57   o 

i5 

0177 

2898 

854o 

838i 

69o3 

3863 

5266 

9346 

3629 

4829 

45 

3o 

oo33 

3n6 

8372 

8636 

6711 

4i55 

5o5o 

9674 

3389 

5i94 

3o 

45 

4.9888 

3334 

8203 

889o 

65i8 

4446 

4832 

S.oooi 

3i47 

5557 

i5 

34  o 

9?4a 

3552 

8o33 

9i44 

6323 

4735 

46i3 

o327 

2904 

59i9 

56   o 

i5 

gSgS 

3768 

7861 

9396 

6127 

5o24 

4393 

o652 

2659 

6280 

,45 

3o 

9448 

3984 

7689 

9648 

593o 

53i2 

4171 

°977 

24i3 

664i 

3o 

45 

9299 

4200 

75i5 

99oo 

5732 

56oo 

3948 

i3oo 

ai65 

7000 

i5 

35   o 

9149 

44i5 

734i 

4-oi5o 

5532 

5886 

3724 

1622 

1915 

7358 

55   o 

i5 

4.8998 

3.4629 

5.7i65 

4-o4oo 

6.533i 

4.6172 

7.3498 

5.i943 

8.i664 

5.7715 

45 

3o 

8847 

4842 

6988 

o649 

5l29 

6456 

3270 

2263 

l4l2 

8070 

3o 

45 

8694 

5o55 

6810 

o897 

4926 

6740 

3o42 

2582 

n57 

8425 

i5 

36   o 

854: 

5267 

663i 

n45 

4721 

7023 

2812 

29OI 

0902 

8779 

54   o 

i5 

8387 

5479 

645  1 

l392 

45i6 

73o5 

258o 

32i8 

o644 

9i3i 

45 

3o 

823i 

5689 

6270 

i638 

43o9 

7586 

2347 

3534 

o386 

9482 

3o 

45 

8o75 

5899 

6088 

i883 

4ioo 

7866 

2Il3 

3849 

OI25 

9832 

i5 

3y   o 

7918 

6109 

59o4 

2I27 

389i 

8i45 

1877 

4i63 

7.9864 

6.0182 

53   o 

i5 

7760 

63i8 

5720 

237I 

368o 

8424 

i64o 

4476 

9600 

0529 

45 

3o 

7601 

6526 

5535 

26i3 

3468 

8701 

1402 

4789 

9335 

0876 

3o 

45 

4.744i 

3.6733 

5.5348 

4-2855 

6.3255 

4.8977 

7.1162 

5.5ioo 

7.9069 

6.1222 

:5 

38   o 

7281 

6940 

5i6i 

3o96 

3o4i 

9253 

092I 

54io 

8801 

i566 

52    O 

i5 

7119 

7146 

4972 

3337 

2825 

9528 

o679 

57i8 

8532 

1909 

45 

3o 

6956 

735i 

4783 

3576 

26o9 

98oi 

o435 

6026 

8261 

225l 

3o 

45 

6793 

V555 

4592 

38i5 

239I 

5.0074 

oi9o 

6333 

7988 

2592 

i5 

3o   o 

6629 

7759 

44oo 

4o52 

2172 

o346 

6.9943 

6639 

77i5 

2932 

5i   o 

i5 

6464 

7962 

4207 

4289 

I95i 

0616 

9695 

6943 

7439 

3271 

45 

3o 

6297 

8i65 

4oi4 

4525 

i73o 

0886 

9446 

7247 

7l62 

36o8 

3o 

45 

6i3i 

8366 

38i9 

476i 

1507 

u55 

9i96 

755o 

6884 

3944 

i5 

4o  o 

5963 

8567 

3623 

4995 

1284 

1423 

8944 

785i 

66o4 

4279 

5o   o 

i5 

4.5794 

3.8767 

5.3426 

4.5229 

6.io59 

5.i69o 

6.869i 

5.8i5i 

7.6323 

6.4612 

45 

3o 

5624 

8967 

3228 

546i 

o832 

i956 

8437 

845o 

6o4i 

4945 

3o 

45 

5454 

9166 

3o3o 

5693 

o6o5 

2221 

8181 

8748 

5756 

5276 

i5 

4i  o 

5283 

9364 

283o 

5924 

o377 

2485 

7924 

9045 

547i 

56o6 

49   o 

i5 

5no 

956i 

2629 

6i54 

0147 

2748 

7666 

934i 

5i84 

5935 

45 

3o 

4937 

9757 

2427 

6383 

5.99i6 

3oio 

7406 

9636 

4896 

6262 

3o 

45 

4763 

9953 

2224 

6612 

9685 

3271 

7i45 

9929 

46o6 

6588 

i5 

42    0 

4589 

4.oi48 

2O20 

6839 

9452 

353o 

6883 

6.O222 

43i4 

69i3 

48   o 

i5 

44i3 

0342 

1816 

7066 

92I7 

3739 

6620 

o5i3 

4022 

7237 

•45 

3o 

4a37 

o535 

i6o9 

729i 

8982 

4047 

6355 

o8o3 

3728 

7559 

3o 

45 

4-4o59 

4.0728 

5.i4o3 

4-75i6 

5.8746 

5.43o4 

6.6o89 

6.1092 

7.3432 

6.7880 

i5 

43  o 

388i 

092O 

n95 

774o 

85o8 

456o 

5822 

i38o 

3i35 

8200 

4?   ° 

i5 

3702 

IIII 

o986 

7963 

8270 

48i5 

5553 

1666 

2837 

85i8 

45 

3o 

3532 

i3oi 

0776 

8i85 

8o3o 

5o68 

5284 

I952 

2537 

8835 

3o 

45 

3342 

i49i 

o565 

84o6 

7789 

532i 

5oi3 

2236 

2236 

9i5i 

i5 

44  o 

3i6o 

1680 

o354 

8626 

7547 

5573 

474i 

2519 

1934 

9466 

46   o 

i5 

2978 

1867 

oi4i 

8845 

73o4 

5823 

4467 

2801 

i63o 

9779 

45 

3o 

2795 

2o55 

4.9928 

9o64 

7060 

6073 

4i93 

3o82 

i325 

7.oo9i 

3o 

45 

2611 

224l 

97i3 

928l 

68i5 

632i 

39i7 

336i 

1019 

o4oi 

i5 

45  o 

2426 

2426 

9497 

9497 

6569 

6569 

364o 

364o 

O7II 

O7II 

45   o 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dist.  6. 

Dist.  7. 

Dist.  8. 

Dist.  9. 

Dist.  10V 

Course. 

142 


MERIDIONAL    PARTS. 


LATITUDE. 

Min. 

0° 

1° 

2° 

3° 

4° 

5° 

6° 

7° 

8° 

9° 

10° 

11° 

12° 

Min. 

0 

o.o 

60.0 

I20.O 

180.1 

240.2 

3oo.2 

360.7 

421. 

48i.6 

542.2 

6o3.i 

664.1 

725.3 

o 

I 

I.O 

61.0 

21.0 

81. 

4i.a 

01.2 

61.7 

22. 

82.6 

43.3 

04.1 

65.3 

26.3 

I 

2 

2.O 

62.0 

22.0 

82. 

42.2 

O2.^ 

62.7 

23. 

83.6 

44.3 

o5.i 

66.1 

27.4 

a 

3 

3.o 

63.o 

23.0 

83. 

43.2 

o3.2 

63.7 

24. 

84-6 

45.3 

06.  i 

67.1 

28.4 

3 

4 

4.0 

64-0 

24.0 

84. 

44-2 

04.^ 

64.7 

25. 

85.6 

46.3 

07.1 

68.2 

29.4 

4 

5 

5.o 

65.o 

25.0 

85. 

45.2 

o5./ 

65.7 

26. 

86.6 

47-3 

08.2 

69.2 

3o.5 

5 

6 

6.0 

66.0 

26.0 

86. 

46.2 

o6./ 

66.7 

27. 

87.6 

48.3 

09.2 

70.2 

3i.5 

6 

7 

7.0 

67.0 

27.0 

87. 

47.2 

07.4 

67.7 

28. 

88.6 

49.3 

10.2 

71.2 

32.5 

7 

8 

8.0 

68.0 

28.0 

88. 

48.2 

08.4 

68.7 

29. 

89.6 

5o.3 

II.  2 

72.2 

33.5 

8 

9 

9.0 

69.0 

29.0 

89. 

49.2 

09.2 

69.7 

3o. 

90.7 

5i.4 

12.2 

73.3 

34.5 

9 

10 

10.  0 

70.0 

iSo.o 

190. 

250.2 

3ioJ 

370.7 

43i. 

491.7 

552.4 

6l3.2 

674.3 

735.6 

10 

ii 

II.  0 

71.0 

3i.o 

91. 

5l.2 

1  1.  L 

71.7 

32. 

92.7 

53.4 

14.2 

75.3 

36.6 

ii 

12 

12.0 

72.0 

32.0 

92. 

52.2 

12.1 

72.7 

33. 

93.7 

544 

i5.3 

76.3 

37.6 

12 

i3 

i3.o 

73.o 

23.o 

93. 

53.2 

I  3.4 

73.7 

34.2 

94.7 

55.4 

i6.3 

77.3 

38.6 

i3 

i4 

i4«o 

74.0 

34.o 

94- 

54.2 

l4-4 

74-7 

35.2 

95.7 

56.4 

i7.3 

78.4 

39.6 

i4 

i5 

i5.o 

75.0 

35.o 

95. 

55.2 

i5.4 

75.7 

36.2 

96.7 

57-4 

i8.3 

79-4 

4o.7 

i5 

16 

16.0 

76.0 

36.o 

96. 

56.2 

16.4 

76.8 

37.2 

97-7 

58.4 

19.3 

80.4 

4i.7 

16 

*7 

17.0 

77.0 

37.0 

97- 

57.2 

i7.5 

77.8 

38.2 

98.7 

59.4 

20.3 

81.4 

42.7 

'7 

18 

18.0 

78.0 

38.o 

98. 

58.2 

iS.5 

78.8 

39.2 

99.8 

6o.5 

21.3 

82.4 

43.7 

18 

'9 

19.0 

79.0 

39.0 

99. 

59.2 

i9.5 

79.8 

40.2 

5oo.8 

6i.5 

22.4 

83.5 

44.8 

'9 

20 

2O.O 

80.0 

i4o.o 

200.  i 

260.2 

320.5 

38o.8 

441.2 

5oi.8 

562.5 

623.4 

684.5 

745.8 

2C 

21 

21.  0 

81.0 

4i.o 

OI.I 

6i.3 

21.5 

81.8 

42.2 

02.8 

63.5 

244 

85.5 

46.8 

21 

22 

22.  0 

82.0 

42.0 

02.1 

62.3 

22.5 

82.8 

43.2 

o3.8 

64-5 

25.4 

86.5 

47.8 

22 

23 

23.0 

83.o 

43.o 

o3.i 

63.3 

23.5 

83.8 

44.2 

o4-8 

65.5 

26.4 

87.5 

48.9 

23 

24 

24.O 

84.0 

44.o 

04.1 

64-3 

24.5 

84-8 

45.2 

o5.8 

66.6 

27.4 

88.6 

49-9 

24 

25 

25.  0 

85.o 

45.o 

o5. 

65.3 

25.5 

85.8 

46.3 

06.8 

67.6 

28.5 

89.6 

5o.9 

25 

26 

26.0 

86.0 

46.o 

06. 

66.3 

26.5 

86.8 

47-3 

07.8 

68.6 

20.5 

90.6 

5i.9 

26 

2? 

27.0 

87.0 

47.0 

07. 

67.3 

27.5 

87.8 

48.3 

08.9 

69.6 

3o.5 

91.6 

53.o 

27 

28 

28.0 

88.0 

48.o 

08. 

68.3 

28.5 

88.8 

49-3 

09.9 

70.6 

3i.5 

92.6 

54.0 

28 

29 

29.0 

89.0 

49.0 

09.1 

69.3 

29.5 

89.8 

5o.3 

10.9 

71.6 

32.5 

93.6 

55.o 

29 

3o 

3o.o 

90.0 

iSo.o 

2IO.I 

270.3 

33o.5 

39o.8 

45i.3 

5  1  1.  9 

572.6 

633.5 

694.7 

756.o 

3o 

3i 

3i.o 

91.0 

5i.o 

II.  I 

7i.3 

3i.5 

91.8 

52.3 

12.9 

73.7 

34.6 

95.7 

57.i 

3i 

32 

32.0 

92.0 

52.  0 

12.  1 

72.3 

32.5 

92.9 

53.3 

i3.9 

74-7 

35.6 

96.7 

58.i 

32 

33 

33.o 

93.0 

53.1 

i3.i 

73.3 

33.5 

93.9 

54.3 

i4-9 

75.7 

36.6 

97:7 

59.i 

33 

34 

34.o 

94.0 

54-1 

14.1 

74.3 

34.5 

94.9 

55.3 

i5.9 

76.7 

37.6 

98.7 

60.1 

34 

35 

35.o 

95.0 

55.i 

i5.i 

75.3 

35.5 

95.9 

56.3 

16.9 

77-7 

38.6 

99.8 

61.1 

35 

36 

36.o 

96.0 

56.i 

16.1 

76.3 

36.5 

96.9 

57.3 

18.0 

78.7 

39.6 

700.8 

62.2 

36 

3? 

37.o 

97.0 

57.i 

17.1 

77-3 

37.5 

97-9 

58.4 

19.0 

79-7 

40.7 

01.8 

63.2 

37 

38 

38.o 

98.0 

58.i 

18.1 

78.3 

38.5 

98.9 

59.4 

20.  o 

80.8 

4i.7 

02.8 

64.2 

38 

39 

39.0 

99.0 

69.1 

19.1 

79.3 

39.6 

99-9 

60.4 

2  I.O 

81.8 

42.7 

o3.8 

65.2 

39 

4o 

4o.o 

IOO.O 

160.1 

220.  2 

280.3 

34o.6 

^00.9 

46i.4 

522.0 

582.8 

643.7 

704.9 

766.3 

4o 

4i 

4i.o 

01.  0 

61.1 

21.2 

8i.3 

4i.6 

01.9 

62.4 

23.0 

83.8 

44-7 

o5.9 

67.3 

4i 

42 

42.0 

02.  o 

62.1 

22.2 

82.3 

42.6 

02.9 

63.4 

24.0 

84.8 

45.8 

06.9 

68.3 

42 

43 

43.o 

o3.o 

63.i 

23.2 

83.3 

43.6 

03.9 

64-4 

25.  0 

85.8 

46.8 

07.9 

69.3 

43 

44 

44.0 

o4.o 

64.i 

24.2 

84.3 

44-6 

o4>9 

65.4 

26.0 

86.8 

47.8 

09.0 

70.4 

44 

45 

45.o 

o5.o 

65.i 

25.2 

85.3 

45.6 

©5.9 

66.4 

27.1 

87.9 

48.8 

IO.O 

7i.4 

45 

46- 

46.o 

06.0 

66.1 

26.2 

86.3 

46.6 

07.0 

67.4 

28.1 

88.9 

49.8 

I  I.O 

72.4 

46 

47 

47-0 

07.0 

67.1 

27.2 

87.3 

47-6 

08.0 

68.4 

29.1 

89.9 

5o.8 

12.0 

73.4 

47 

48 

48.o 

08.0 

68.1 

28.2 

88.3 

48.6 

09.0 

69.5 

3o.i 

90.9 

5i.9 

i3.i 

74.5 

48 

49 

49-0 

09.0 

69.1 

29.2 

89.3 

49.6 

10.0 

70.5 

3i.i 

91.9 

52.  9 

14.1 

75.5 

49 

5o 

5o.o 

IIO.O 

170.1 

230.2 

290.3 

35o.6 

4n.o 

471.5 

532.1 

592.9 

653.9 

7i5.i 

776.5 

5o 

5i 

5i.o 

II.  0 

71.1 

3l.2 

9i.3 

5i.6 

12.  0 

72.5 

33.1 

93.9 

54-9 

16.1 

77.5 

5i 

5  2 

52.  0 

12.0 

72.1 

32.2 

92.4 

52.6 

i3.o 

73.5 

34-1 

95.0 

55.9 

17.1 

78.6 

52 

53 

53.o 

i3.o 

73.i 

33.2 

93.4 

53.6 

i4-o 

74-5 

35.i 

96.0 

57.o 

18.2 

79.6 

53 

54 

54.o 

14.0 

74.1 

34.2 

944 

54.6 

i5.o 

75.5 

36.2 

97.0 

58.o 

19.2 

80.6 

54 

55 

55.o 

i5.o 

75.i 

35.2 

95.4 

55.6 

16.0 

76  5 

37.2 

98.0 

59.0 

20.2 

81.7 

55 

56 

56.o 

16.0 

76.1 

36.2 

96.4 

56.6 

17.0 

77.5 

38.2 

99.0 

60.0 

21.2 

82.7 

56 

57 

57.0 

17.0 

77.1 

37.2 

97-4 

57.6 

18.0 

78.5 

39.2 

600.0 

61.0 

22.3 

83.7 

57 

58 

58.o 

18.0 

78.1 

38.2 

9B-4 

58.6 

19.0 

79.5 

4O.2 

01.  0 

62.1 

23.3 

84-7 

58 

59 

Sg.o 

19.0 

79.1 

39.2 

99.4 

59.7 

20.  o 

8o.5 

41.2. 

02.  1 

63.i 

24-3 

85.8 

59 

MERIDIONAL    PARTS. 


143 


LATITUDE. 

1 

13° 

14° 

15° 

16° 

17° 

18° 

19° 

20° 

21° 

22° 

23° 

24° 

.a 

53 

o 

786.8 

848.5 

910.5 

972.7 

io35.3 

1098.2 

u6i.5 

1225.  1 

I289.2 

i353.7 

i4i8.6 

i484-i 

o 

i 

87.8 

49.5 

i.i-.5 

73.8 

36.3 

99.3 

62.5 

26.2 

9o.3 

54.8 

i9.7 

85.2 

I 

2 

88.8 

5o.5 

12.6 

74-8 

374 

noo.3 

63.6 

27.3 

9i.3 

55.8 

20.8 

86.3 

2 

3 

89.9 

5i.6 

i3.6 

75.9 

38.4 

01.4 

64-7 

28.3 

92.4 

56-9 

21.9 

87.3 

3 

4 

90.9 

52.6 

i4-6 

76.9 

39.5 

02.4 

65.7 

29.4 

93.5 

58.o 

23.0 

88.4 

4 

5 

91.9 

53.6 

i5.7 

78.0 

4o.5 

o3.5 

66.8 

3o.4 

94.5 

Sg.o 

24.1 

89.5 

5 

6 

92.9 

54.7 

16.7 

79.0 

4i.6 

o4.5 

67.8 

3i.5 

95.6 

60.  i 

25-1 

9o.6 

ti 

7 

94.0 

55.7 

17.7 

80.0 

42.6 

o5.6 

68.9 

32.6 

96-7 

61.2 

26-2 

91.7 

7 

8 

95.0 

56.7 

18.8 

81.1 

43.7 

06.6 

70.0 

33.6 

97-8 

62.3 

27.3 

92.8 

8 

9 

96.0 

57.8 

19.8 

82.1 

44-7 

07.7 

71.0 

34.7 

98.8 

63.4 

28.4 

93.9 

9 

10 

797.0 

858.8 

920.8 

983.2 

io45.8 

1108.7 

1172.1 

1235.8 

I299.9 

i364.5 

i429.5 

i495.o 

10 

ii 

98.1 

59.8 

21.9 

84.2 

46.8 

09.8 

73.i 

36.8 

iSoi.o 

65.6 

3o.6 

96.i 

ii 

12 

99.1 

60.9 

22.9 

85.2 

47-9 

10.8 

74.2 

37.9 

O2.O 

66.6 

3i.7 

97-2 

12 

i3 

800.1 

61.9 

23.9 

86.3 

48.9 

11.9 

75.2 

39.0 

o3.i 

67.7 

32.8 

98.3 

i3 

i4 

OI.2 

62.9 

25.  0 

87.3 

49.9 

12.9 

76.3 

4o.o 

04.2 

68.8 

33.9 

99-4 

i4 

i5 

O2.  2 

64-o 

26.0 

88.4 

5i.o 

14.0 

77-4 

4i.i 

o5.3 

69.9 

34.9 

i5oo.5 

i5 

16 

03.2 

65.o 

27.0 

89.4 

52.0 

i5.o 

78.4 

42.2 

o6.3 

70.9 

36.o 

01.6 

16 

!? 

04«2 

66.0 

28.1 

90.4 

53,1 

16.1 

79.5 

43.2 

07.4 

72.0 

87.1 

02.7 

«7 

18 

o5.3 

67.1 

29.1 

9i.5 

54-1 

17.1 

8o.5 

44-3 

o8.5 

73.i 

38.2 

o3.8 

18 

J9 

06.3 

68.1 

3o.i 

92.5 

55.2 

18.2 

81.6 

45.4 

o9.6 

74-2 

39.3 

o4-9 

J9 

20 

807.3 

869.1 

93l.2 

993.6 

io56.2 

1119.2 

1182.7 

1246.4 

i3io.6 

i375.3 

i44o.4 

i5o6.o 

20 

21 

08.4 

70.1 

32.2 

94-6 

57.3 

20.3 

83.7 

47-5 

11.7 

76.4 

4i.5 

07.1 

21 

22 

09.4 

71.2 

33.3 

95.6 

58.3 

21.3 

84.8 

48.6 

12.8 

77-4 

42.6 

08.2 

22 

23 

10.4 

72.2 

34-3 

96-7 

69.4 

22.4 

85.8 

49.6 

i3.8 

78.5 

43.7 

09.3 

23 

24 

11.4 

^3.2 

35.3 

97-7 

60.4 

23.4 

86.9 

5o.7 

i4.9 

79-6 

44-8 

10.4 

24 

25 

12.5 

"4.3 

36.3 

98.8 

6i.4 

24.5 

88.0 

5i.8 

16.0 

80.7 

45.8 

n.  5 

26 

26 

i3.5 

75.3 

374 

99.8 

62.5 

25.5 

89.o 

62.8 

17.1 

81.8 

46.9 

12.6 

26 

27 

i4.5 

76.3 

38.4 

1000.8 

63.5 

26.6 

9o.i 

53.9 

18.1 

82.8 

48.o 

i3.7 

27 

28 

i5.5 

77-4 

39.5 

oi.9 

64.6 

27.6 

91.1 

55.o 

19.2 

83.9 

49.i 

i4.8 

28 

29 

16.6 

78.4 

4o.5 

02.  9 

65.6 

28.7 

92.2 

56.o 

20.3 

,85.o 

50.2 

i5.9 

29 

3o 

817.6 

8794 

94i.6 

ioo4«o 

1066.7 

1129.7 

II93.2 

1257.1 

i32i.4 

i386.i 

i45i.3 

1517.0 

3o 

3i 

18.6 

8o.5 

42.6 

o5.o 

67.7 

3o.8 

94-3 

58.2 

22.5 

87.2 

52.4 

18.1 

3i 

32 

19.6 

8i.5 

43.6 

06.  i 

68.8 

3i.8 

95.4 

59.2 

23.5 

88.3 

53.5 

19.2 

32 

33 

20.7 

82.5 

44-7 

07.1 

69.8 

32.9 

96.4 

6o.3 

24.6 

89.^ 

54.6 

20.^ 

33 

34 

21.7 

83.6 

45-7 

08.  i 

70.9 

34.o 

97.5 

61.4 

25.7 

90.4 

55.6 

21.^ 

34 

35 

22.7 

84.6 

46.7 

09.2 

72.0 

35.1 

98.5 

62.4 

26.7 

9i.5 

66.7 

22.5 

35 

36 

23.8 

85.6 

47-8 

IO.2 

73.0 

36.i 

99.6 

63.5 

27.8 

92.6 

57.8 

23.6 

36 

3? 

24.8 

86.7 

48.8 

II.  3 

74.1 

37.2 

1200.7 

64-6 

28.9 

93.7 

58.9 

24.7 

37 

38 

25.8 

87.7 

49.9 

12.3 

75.i 

38.2 

01.7 

65.6 

3o.o 

94.8 

60.0 

25.8 

38 

39 

26.9 

88.7 

5o.9 

1  3.4 

76.2 

39.3 

02.  8 

66.7 

3i.i 

95.8 

61.1 

26.9 

39 

4o 

827.9 

889.8 

961.9 

ioi4-4 

1077.2 

n4o.3 

I2o3.9 

1267.8 

i332.i 

i396.9 

:462.2 

1  528.0 

4o 

4i 

28.9 

90.8 

53.o 

1  5.4 

78.3 

4i4 

04.9 

68.8 

33.2 

98.o 

63.3 

29.1 

4i 

42 

29.9 

91.8 

54.o 

i6.5 

79.3 

42.4 

06.0 

69.9 

34.3 

99.i 

644 

30.2 

42 

43 

3i.o 

92.9 

55.i 

i7.5 

80.4 

43.5 

07.1 

71.0 

35.3 

I4OO.2 

65.5 

3i.3 

43 

44 

32.0 

93.9 

56.i 

18.6 

81.4 

44.6 

08.  i 

72.1 

36.4 

OI.£ 

66.6 

32.4 

44 

45 

33.o 

94.9 

57.i 

19.6 

82.5 

45.6 

09.2 

73.i 

37.5 

O2./! 

67.7 

33.5 

45 

46 

34.i 

96.0 

58.2 

2O.6 

83.5 

46.7 

IO.2 

74.2 

38.6 

o3.4 

68.8 

34.6 

46 

4? 

35.i 

97.0 

59.2 

21.7 

84-6 

47-7 

ii.  3 

75.3 

39.7 

o4-5 

69.8 

3/>.7 

4? 

48 

36.i 

98.0 

60.2 

22.7 

85.6 

48.8 

12.4 

76.3 

4o.7 

o5.6 

7°-9 

36.8 

48 

49 

37.2 

99.1 

6i.3 

23.8 

86.7 

49.9 

1  3.4 

77-4 

4i.8 

06.7 

72.0 

37.9 

49 

5o 

838.2 

900.1 

962.2 

1024.8 

1087.7 

n5o.9 

i2i4.5 

!278.5 

1342.9 

1407.8 

i473.i 

iSSg.o 

5o 

5i 

3g.2 

01.  1 

63.4 

25.9 

88.8 

52.0 

i5.5 

79.5 

44-0 

08.8 

74.2 

4o.i 

5i 

62 

4O.2 

O2.  2 

64.4 

26.9 

89.8 

53.o 

16.6 

80.6 

45.i 

°9!9 

75.3 

4i.a 

5s 

53 

4i.3 

03.2 

65.5 

28.0 

9°  9 

54.i 

17.7 

81.7 

46.i 

II.  O 

76.4 

42.3 

53 

54 

42.3 

o4.3 

66.5 

29.0 

91.9 

55.i 

18.7 

82.8 

47.2 

12.  1 

77.5 

43.4 

54 

55 

43.3 

o5.3 

67.5 

3o.i 

93.0 

56.2 

19.8 

83.8 

48.3 

13.2 

78.6 

44-5 

55 

56 

444 

o6.3 

68.6 

Sz.i 

94.0 

57.2 

20.9 

84-9 

49.4 

i4-3 

79-7 

45.6 

56 

57 

45.4 

07-4 

69.6 

32.2 

95.i   58.3 

21.9 

86.0 

5o.4 

1  5.4 

80.8 

46.7 

57 

58 

46-4 

08.^ 

70.7 

33.2 

96.1 

59.4 

23.0 

87.0 

5i.5 

i6.5 

8i.9 

47-8 

58 

59 

47-5 

09.  i 

71.7 

34.3 

97.2 

60.4 

24.1 

88.1 

52.6 

i7.5 

83.o 

46-9 

59 

M  E  ii  i  D  i  o  N  A  L    PARTS. 


LATITUDE. 

MlE. 

25° 

26° 

27° 

28° 

29° 

30° 

31° 

32° 

33° 

54° 

35° 

Min. 

0 

i55o.o 

i6i6.5 

!683.5 

I75l.2 

i8i94 

1888.4 

i958.o 

2028.4 

200,  0,.  5 

2171.5 

2244.3 

0 

I 

5i.i 

17.6 

84.6 

52.3 

2O.6 

89.5 

59.2 

29.6 

2100.7 

72.7 

45.5 

I 

2 

52.2 

18.7 

85.8 

534 

21.7 

90.7 

60.4 

3o.7 

01.9 

73.9 

46.8 

2 

3 

53.3 

i9.8 

86.9 

54.6 

22.  9 

91.9 

61.6 

3r.9 

03.! 

75.i 

48.o 

3 

4 

544 

2O.9 

88.0 

55.7 

24.O 

93.o 

62.7 

33.i 

o4.3 

76.3 

49.2 

4 

5 

55.5 

22.  0 

89.1 

56.8 

25.2 

94-i 

63.9 

34.3 

o5.5 

77.5 

5o4 

5 

6 

56.6 

23.2 

90.3 

58.o 

26.3 

95.3 

65.o 

35.5 

06.7 

78.7 

5i.6 

6 

7 

57.7 

24.3 

91.4 

59.i 

27.5 

96.5 

66.2 

36.7 

07.9 

80.0 

52.  9 

7 

8 

58.8 

254 

92.5 

60.2 

28.6 

97-6 

674 

37.8 

09.1 

81.2 

54.1 

8 

9 

59.9 

26.5 

93.6 

61.4 

29'7 

98.8 

68.5 

39.o 

io.3 

82.4 

55.3 

9 

10 

i56i.o 

1627.6 

1694.8 

1762.5 

i83o.9 

i899.9 

i969.7 

2O4O.2 

2III.5 

2i83.6 

2256.5 

10 

ii 

62.1 

28.7 

95.9 

63.6 

32.0 

70-9 

4x4 

12.7 

84.8 

57.8 

i 

12 

63.2 

20.8 

9-7.0 

64.8 

33.2 

02.3 

72.0 

42.6 

13-9 

86.0 

59.o 

12 

r3 

644 

3i.o 

98.1 

65.9 

34-3 

o34 

73.2 

43.8 

87.2 

60.2 

i3 

i4 

65.5 

32.1 

99.3 

67.0 

35.5 

o4-6 

744 

44.9 

i6.3 

884 

61.4 

i4 

i5 

66.6 

33.2 

I700.4 

68.2 

36.6 

o5.7 

75.6 

46.i 

i7.5 

89.6 

62.  7 

i5 

16 

67.7 

34.3 

oi.5 

69.3 

37.8 

06.  9 

76.8 

47.3 

18.7 

9o.8 

63.9 

16 

'7 

68.8 

35.4 

02.6 

70.5 

38.9 

08.1 

77-9 

48.5 

19.8 

02.  o 

65.i 

i7 

18 

69.9 

36.5 

o3.8 

71.6 

4o.i 

O9.2 

79.i 

49-7 

21.0 

93.3 

66.3 

18 

J9 

71.0 

37.6 

o4-9 

72.7 

41.2 

IO.4 

80.2 

5o.8 

22.2 

944 

67.5 

J9 

20 

1572.1 

1638.8 

I7o6.o 

1773.9 

1842.4 

i9n.5 

i98i.4 

2O52.O 

21234 

2195.7 

2268.8 

20 

21 

73.2 

39.9 

O7.l 

75.0 

43.5 

12.7 

82.6 

53.2 

24.6 

96.9 

7o.o 

21 

22 

74.3 

4i.o 

o8.3 

76.1 

44.6 

i3.8 

83.7 

544 

25.8 

98.1 

71.2 

22 

23 

754 

42.1 

o94 

77.2 

45.8 

i5.o 

84.0. 

55.6 

27.0 

99.3 

72.5 

23 

24 

76.5 

43.2 

io.5 

784 

46.9 

16.2 

86.1 

56.8 

28.2 

220O.5 

73.7 

24 

25 

77.6 

44-3 

ii.  6 

79.5 

48.i 

i7.3 

87.3 

58.o 

20.4 

01.7 

25 

26 

78.7 

45.5 

12.8 

80.6 

49.2 

i8.5 

884 

59.i 

3o.6 

o3.o 

76.i 

26 

27 

79-8 

46.6 

i3.9 

81.8 

5o4 

i9.6 

89.6 

6o.3 

3i.8 

04.2 

774 

27 

28 

8o.9 

47-7 

i5.o 

83.o 

5i.5 

20.8 

9o.8 

6i.5 

33.o 

o54 

78.6 

28 

29 

82.1 

48.8 

16.1 

84.1 

52.7 

21.  9 

92.O 

62.  7 

34.2 

06.6 

79-8 

29 

3o 

i583.2 

i649.9 

I7I7-3 

1785.2 

i853.8 

I923.I 

!993.I 

2o63.9 

2i354 

2207.8 

2281.0 

3o 

3r 

84.3 

5i.o 

18.4 

864 

55.o 

24.3 

94.3 

65.i 

36.6 

09.0 

82.3 

3i 

32 

854 

52.2 

I9.5 

87.5 

56.i 

254 

95.5 

66.2 

37.8 

10.2 

83.5 

32 

33 

86.5 

53.3 

20.  7 

88.6 

57.2 

26.6 

96.6 

674 

39.o 

II.4 

84-7 

33 

34 

87.6 

544 

21.8 

89.8 

584 

27.8 

97-8 

68.6 

4o.2 

12-7 

86.0 

34 

35 

88.7 

55.5 

22.  9 

9°-9 

59.6 

28.0 

99-° 

69.8 

4i4 

1  3.9 

87.2 

35 

36 

89.8 

56.6 

24.0 

92.1 

60.  7 

3o.i 

2OOO.2 

71.0 

42.6 

i5.i 

884 

36 

37 

9°-9 

57.8 

25.2 

93.2 

61.9 

3i.3 

01.3 

72.2 

43.8 

i6.3 

89-7 

37 

38 

92.O 

58.9 

26.3 

94.3 

63.o 

32.4 

02.  5 

734 

45.o 

i7.5 

9°-9 

38 

39 

93.: 

60.0 

274 

95.5 

64-2 

33.6 

o3.7 

74-5 

46.2 

18.7 

92.1 

39 

4o 

i594.3 

1661.1 

1-728.6 

i796.6 

i865.3 

i934.7 

2OO4.9 

2075.7 

2147.4 

2219.9 

2293.3 

4o 

4i 

96.4 

62.2 

29.7 

97-8 

66.5 

35.9 

O6.O 

76.9 

48.6 

21.2 

94.6 

4i 

42 

96.5 

634 

3o.8 

98.9 

67.6 

37.i 

07.2 

78.1 

49.8 

22.4 

95.8 

42 

43 

97«6 

64.5 

3i.9 

1800.0 

68.8 

38.2 

084 

79.3 

5i.o 

23.6 

97.0 

43 

44 

98.7 

65.6 

33.i 

01.2 

69.9 

394 

o9.6 

8o.5 

52.2 

24.8 

98.3 

44 

45 

99-8 

66.7 

34.2 

02.3 

7i.i 

4o.5 

10.7 

81.7 

53.4 

26.0 

99.5 

45 

46 

i6oo.9 

67.8 

35.3 

o3.5 

72.2 

4i.7 

11.9 

82.0 

54.6 

27.2 

23oo.7 

46 

47 

02.  o 

69.o 

36.5 

o4.6 

734 

42.  0 

84.o 

55.8 

28.5 

02.  0 

47 

48 

o3.i 

70.1 

37.6 

05.7 

74.5 

44.o 

i4^3 

85.2 

57.0 

29.7 

03.2 

48 

49 

04.2 

71.2 

38.7 

o6.9 

75.7 

45.2 

1  5.4 

864 

58.2 

30.9 

o44 

49 

5o 

i6o54 

i672.3 

I739.9 

1808.0 

i876.8 

i9464 

2016.6 

2087.6 

2i594 

2232.1 

23o5.7 

5o 

5i 

o6.5 

734 

4i.o 

09.2 

78.o 

47.5 

17.8 

88.8 

60.7 

33.3 

06.9 

5i 

52 

07.6 

74-5 

42.1 

io.3 

•79.2 

48.7 

I9.0 

90.0 

6i.9 

34-6 

08.  1 

53 

53 

08.7 

75.7 

43.2 

11.4 

8o.3 

49.9 

20.2 

91.2 

63.i 

35.8 

09.4 

53 

54 

o9.8 

76.8 

444 

12.6 

8i.5 

5i.o 

21.3 

92.4 

64.3 

37.0 

10.6 

54 

55 

10.  9 

77-9 

45.5 

i3.7 

82.6 

52.2 

22.5 

93.6 

65.5 

38.2 

11.8 

55 

56 

12.  0 

79-° 

46.6 

I4.9 

83.8 

534 

23.7 

94.8 

66.7 

394 

i3.i 

56 

57 

i3.i 

80.2 

47-8 

16.0 

84.9 

54.5 

96.0 

67-9 

4o.7 

i4.3 

57 

58 

14.2 

8i.3 

48.9 

17.2 

86.1 

55.7 

26.0 

97.1 

69-i 

41.9 

i5.5 

58 

59 

i54 

82.4 

5o.o 

i8.3 

87.2 

56.9 

27.2 

98.3 

70.3 

43.i 

1  6.8 

59 

MERIDIONAL    PARTS. 


145 


LATITUDE. 


1  MiD  |T00 

37° 

38° 

39° 

40° 

41° 

42° 

43° 

44° 

45° 

46° 

m, 

o  I23i8.o 

2392.6 

2468.3 

2545.o 

2622.7 

2701.6 

2781.7 

2863.1 

2945.  8 

3o3o.o 

3n5.6 

O 

I 

19.2 

93.9 

69.5 

46.2 

24.0 

02.9 

83.i 

64.5 

47.2 

3i.4 

17.0 

I 

2 

20.5 

95.i 

70.8 

47-5 

25.3 

o4-3 

84-4 

65.8 

48.6 

32.8 

i8.5 

2 

3 

21.  7 

96.4 

72.1 

48.8 

26.6 

o5.6 

85.8 

67.2 

5o.o 

34.2 

i9.9 

3 

4 

23.0 

97-7 

73.4 

5o.i 

27.9 

06.9 

87.1 

68.5 

5i.4 

35.6 

21.4 

4 

5 

24.2 

98.9 

74.6 

5i.4 

29.2 

o8.3 

88.5 

70.0 

52.8 

37.o 

22.8 

5 

6 

25.4 

2400.2 

75.9 

52.7 

3o.5 

09.6 

89.8 

7i.3 

54-2 

38.4 

24.2 

6 

7 

26.7 

01.4 

77-i 

54.o 

31.9 

10.9 

91.2 

72.7 

55.6 

39.8 

25.7 

7 

8 

2-7.9 

02.7 

78.5 

55.3 

33.2 

12.2 

92.5 

74.1 

57.0 

4i.3 

27.1 

8 

9 

29.1 

03.9 

79-7 

56.6 

34.5 

i3.5 

93.8 

75.4 

58.3 

42.7 

28.5 

9 

10 

233o.4 

24o5.2 

2481.0 

2557.8 

2635.8 

2714.9 

2795.i 

2876.8 

2959.8 

3o44.i 

3i3o.o 

10 

ii 

3i.6 

06.4 

82.2 

59.i 

37.i 

I  6.2 

96.5 

78.2 

61.1 

45.5 

3i.5 

n 

12 

32.9 

07.7 

83.5 

60.4 

38.4 

17.5 

97-9 

79.5 

62.5 

47.0 

32.9 

12 

i3 

34.i 

09.0 

84-8 

61.7 

39.7 

18.9 

99.3 

80.9 

63.9 

48.4 

34.3 

i3 

i4 

35.3 

10.2 

86.1 

63.o 

4i.o 

20.2 

2800.6 

82.3 

65.3 

49-8 

35.8 

i4 

i5 

36.6 

ii.  5 

87.4 

64-3 

42.3 

21.5 

02.  o 

83.7 

66.7 

5i.? 

37.2 

i5 

16 

37.8 

12.7 

88.6 

65.6 

43.6 

22.9 

o3.3 

85.o 

68.1 

52.6 

38.7 

16 

J7 

39.o 

i4«o 

89.9 

66.9 

44.9 

24-2 

04.7 

86.4 

6c;.5 

54-1 

4o.i 

»7 

18 

4o.3 

15.2 

91.2 

68.2 

46.3 

25.5 

06.0 

87.8 

70.9 

55.5 

4i.6 

18 

»9 

4i.5 

i6.5 

92.4 

69.5 

47-6 

26.8 

o7.3 

89.1 

72.3 

56.9 

43.o 

'9 

20 

2342.8 

2417-8 

2493.7 

257o.7 

2648.9 

2728.212808.8 

2890.5 

2973.7 

3o58.3 

3i44-5 

20 

21 

44.o 

19.0 

95.0 

72.0 

5o.a 

29.5 

10.  1 

91.9 

75.i 

59.7 

45-9 

21 

22 

45.3 

20.3 

96.3 

73.3 

5i.5 

3o.8 

11.4 

93.3 

76.5 

61.2 

47-4 

22 

23 

46.5 

21.5 

97.6 

74-6 

52.8 

02.2 

12.8 

94-7 

77-9 

62.6 

48.8 

23 

24 

47-7 

22.8 

98.8 

75.9 

54-1 

33.5 

14.1 

96.0 

.79.3 

64-0 

5o.3 

24 

25 

49.0 

24.0 

25oo.i 

77.2 

55.5 

34.8 

i5.5 

97«4 

80.7 

65.4 

5i.7 

25 

26 

5o.2 

25.3 

01.4 

78.5 

56.8 

36.2 

16.8 

98.8 

82.1 

66.9 

53.2 

26 

27 

5j.5 

26.5 

02.7 

79.8 

58.i 

37.5 

18.2 

2900.2 

83.5 

68.3 

54-6 

27 

28 

52.  7 

27.8 

03.9 

81.1 

59.4 

38.8 

i9.5 

oi.5 

84.9 

69.7 

56.i 

28 

29 

54.0 

29.1 

05.2 

82.4 

66.7 

40.2 

20.9 

02.9 

86.3 

7-1.1 

57.5 

29 

3c 

2355.2 

243o.3 

25o6.5 

a583.7 

2662.0 

2741.5 

2822.3 

2904.3 

2987.7 

3072.6 

SiSg.o 

3o 

3i 

56.5 

3i.6 

07.8 

85.o 

63.3 

42.9 

23.6 

05.7 

89.1 

74.o 

60.4 

3i 

32 

57.7 

32.9 

09.0 

86.3 

64-6 

44-2 

25.  0 

07.1 

90.5 

75.4 

61.9 

32 

33 

58.9 

34-1 

io.3 

87.6 

66.0 

45.5 

26.3 

08.4 

91.9 

76.9 

63.3 

33 

34 

60.2 

35.4 

n.6 

88.9 

67.3 

46.9 

27.7 

09.7 

93.3 

78.3 

64-8 

34 

35 

6i.4 

36-7 

12.9 

90.2 

68.6 

48.2 

29.0 

II.  2 

94-7 

79-7 

66.2 

35 

36 

62  7 

37.9 

14.2 

9i.5 

69.9 

49.5 

3o.4 

•12.6 

96.i 

81.1 

67.7 

36 

37 

63.9 

39.2 

16.4 

92.8 

71.2 

So.g 

3i.7 

i4-o 

97.5 

82.6 

69.1 

37 

38 

65.2 

4o.4 

16.7 

94.i 

72.5 

52.2 

33.i 

i5.3 

98.9 

84.0 

7o.6 

38 

39 

66.4 

4i.7 

18.0 

95.4 

73.9 

53.5 

34-5 

16.7 

3ooo.3 

85.4 

72.O 

39 

4o 

2367.6 

2443.0 

2519.3 

2596.7 

2675.2 

2754.9 

2835.8 

2918.1 

3ooi.8 

3086.9 

3i73.5 

4o 

4i 

68.9 

44-2 

2O.5 

98.o 

76.5 

56.2 

37.2 

19.5 

03.2 

88.3 

75.o 

4i 

42 

7O.2 

45.5 

21.8 

99-3 

77.8 

57.6 

38.6 

20.9 

o4,6 

89.7 

76.4 

42 

43 

71.4 

46.8, 

23.1 

2600.5 

79.1 

58.9 

3y.9 

22.3 

06.0 

91.2 

77-9 

43 

44 

72.6 

48.o 

24.4 

oi.9 

8o.5 

60.2 

4i.3 

23.6 

07.4 

92.6 

79.3 

44 

45 

73-9 

49.3 

25.7 

03.2 

81.8 

6i.5 

42.6 

25  0 

08.8 

94.o 

80.8 

45 

46 

75.i 

5o.6 

27.0 

o4.5 

83.i 

62.9 

44.o 

26.4 

IO.2 

95.5 

82.3 

46 

47 

76.4 

5i.8 

28.3 

o5.8 

84-4 

64.3 

45.4 

27.8 

n.6 

96.9 

83.7 

47 

48 

77.6 

53.i 

29.5 

07.1 

85.7 

65.6 

46.7 

29.2 

i3.o 

98.3 

85.2 

48 

49 

78.9 

54-3 

3o.8 

08.4 

87.1 

66.9 

48.i 

3o.6 

i44 

99-7 

86.6 

49 

5o 

238o.i 

2455.6 

2532.1 

26o9.7 

2688.4 

2768.3 

2849.5 

2932.0 

3oi5.8 

3lOI.2 

3i88.i 

5o 

5i 

81.4 

56.9 

33.4 

II.  0 

89.7 

69.6 

5o.8 

33.3 

17.2 

02.  6 

89.6 

5i 

52 

82.6 

58.i 

34.7 

12.3 

91.0 

71.0 

52.2 

34-7 

18.7 

04.1 

91.0 

52 

53 

83.9 

59.4 

36.o 

i3.6 

02.3 

72.3 

53.5 

36.i 

20.  i 

o5.6 

92.5 

53 

54 

85.i 

60.7 

37.2 

i4-9 

93.7 

73.7 

54-9 

37.5 

21.5 

07.0 

94.0 

54 

55 

86.4 

61.9 

38.5 

16.2 

95.0 

75.0 

56.3 

38.9 

22.  9 

o8.4 

95.4 

55 

56 

87.6 

63.2 

39.8 

i7.5   96.3 

76.3 

57.7 

4o.3 

24-3 

09.8 

96.9 

56 

5? 

88.9 

64.5 

4i.i 

18.8 

9.7.6 

77-7 

5g.o 

4i-7 

25.7 

II.  2 

98.4  57 

58 

90.2 

65.8 

42-4 

20.1 

99.0 

79.0 

6o.5 

43.i   27.1 

12.7 

99  8  58 

69 

91.4 

67.0 

43.6 

21.4 

2700.3 

80.4 

61.7 

44.4   2^.5,  14.1  8201.3  5) 

K 


MERIDIONAL    PARTS. 


LATITUDE. 


Min. 

47° 

48' 

49° 

50° 

51° 

52° 

53° 

54° 

55° 

56° 

57° 

0 

3202.7 

3291.5 

3382.1 

3474.5 

3568.8 

3665.2 

3763.8 

3864.6 

3968.0  4o73.q 

4182  6 

I 

o4«2 

93.0 

83.6 

76.0 

70.4 

66.8 

65.4 

66.3 

69.7 

vS.1- 

84.5 

2 

©5.7 

94.5 

85.i 

77.6 

72.0 

b8.4 

67.1 

68.0 

7i.5 

77- 

86.3 

3 

07.1 

96.0 

86.7 

79-i 

73.6 

70.1 

68.8 

69.7 

73.2 

79.3 

88.1 

A 

08.6 

97.5 

88.2 

80.7 

75.2 

71.7 

70.4 

71.5 

75.o 

8l.T 

90.0 

5 

I  O.I 

99.0 

89.7 

82.3 

76.8 

73.3 

72.1 

73.2 

76.7 

82.9 

91.8 

6 

ii.  5 

33oo.5 

9i.3 

83.8 

78.4 

75.o 

73.7 

74.9 

784 

84.7 

;3-7 

7 

i3.o 

02.0 

92.8 

85.4 

79-9 

76.6 

75.4 

76.6 

80.2 

86.4 

95.5 

8 

14.5 

o3.5 

94.3 

87.0 

8i.5 

78.2 

77.1 

78.3 

82.0 

88.2 

97-3 

9 

i5.9 

OD.O 

95.8 

88.5 

83.i 

79.8 

78.7 

80.0 

83.7 

90.0 

99-2 

10 

3217.4 

33o6.5 

3397.4 

349o.i 

3584-7 

368i.5 

3780.4 

388i.7 

3985.4 

4091.8 

4201.0 

ii 

18.9 

08.0 

98.9 

91.6 

86.3 

83.i 

82.1 

83.4 

87.2 

93.6 

02.  9 

12 

2O.3 

o9.5 

34oo.4 

93.2 

87.9 

84-7 

83.7 

85.i 

88.9 

95.4 

04.7 

i3 

21.8 

II.  0 

02.0 

94.7 

89.5 

86.4 

85.4 

86.8 

9°-7 

97-2 

06.6 

i4 

23.3 

12.5 

o3.5 

96.3 

91.1 

88.0 

87.1 

88.5 

'   L 
92.5 

99-° 

08.4 

i5 

24.8 

i4-o 

o5.o 

97-9 

92.7 

89.6 

88.8 

90.2 

94.2 

4ioo.8 

io.3 

16 

26.2 

i5.5 

06.6 

99.4 

94-3 

9i.3 

9o.4 

92.0 

96.o 

02.6 

12.  1 

I7 

27.7 

17.0 

08.  i 

35oi.o 

95.9 

92.9 

92.1 

93-7 

97-7 

o44 

14.0 

18 

29.2 

i8.5 

o9.6 

02.6 

97-5 

94.5 

93.8 

95.4 

99.5 

06.2 

1.5.8 

19 

3o.7 

20.O 

ii.  i 

04.1 

99.i 

96.2 

95.5 

4001.2 

08.0 

17.7 

20 

3232.1 

3321.5 

3412.7 

35o5.7 

3600.7 

3697.8 

3797.i 

3898.8 

4oo3.o 

4io9.8 

4219.5 

21 

33.6 

23.0 

14.2 

o7.3 

02.  3 

99-4 

98.8 

39oo.5 

04.7 

11.6   21.4 

22 

35.i 

24.5 

i5.7 

08.8 

o3.9 

37oi.i 

38oo.5 

O2.  2 

06.5 

i3.4 

23.2 

23 

36.6 

26.0 

i7.3 

10.4 

o5.5 

02.  7 

02.2 

o4.o 

08.3 

15.2 

25  I 

24 

38.o 

27.5 

18.8 

12.0 

07.1 

o44 

o3.8 

o5.7 

10.  0 

17.1 

27.0' 

25 

39.5 

29.0 

20.4 

i3.5 

08.7 

06.0 

-o5.5 

07.4 

ii.  8 

18.9 

28.8 

26 

4i.o 

3o.6 

21.9 

i5.i 

io.3 

07.6 

07.2 

o9.i 

i3.5 

20.7 

3o.6 

27 

42.5 

32.1 

23.5 

16.7 

n.9 

o9.3 

08.  9 

10.8 

i5.3 

22.5 

32.5 

28 

44.o 

33.6 

25.0 

i8.3 

i3.6 

1  0.0 

io.5 

12.5 

17.1 

24-3 

34.4 

29 

45.4 

35.i 

26.5 

19.8 

i.5.i 

12.6 

12.2 

i4-3 

18.8 

26.1 

36.2 

3o 

3246.9 

3336.6 

3428.0 

3521.4 

3616.7 

3714.2 

38i3.9 

39i6.p 

4020.6 

4127.9 

4238.1 

5i 

48-4 

38.i 

29.6 

23.0 

18.4 

i5.8 

10.6 

17.7 

22.4 

29.7 

4o.o 

32 

49.9 

39.6 

3i.i 

24.6 

20.  o 

i7.5 

i7.3 

19.4 

24-1 

3i.5 

4i.8 

33 

5i.4 

4i.i 

32.7 

26.1 

21.6 

18.9 

21.2 

25.9 

33.3 

43.7 

34 

52.8 

42.6 

34-2 

27.7 

23.2 

20.  8 

20.6 

22.  9 

27.7 

35.2 

45.5 

35 

54.3 

44.i 

35.8 

29.3 

24.8 

22.4 

22.3 

24-6 

29.4 

37.0 

47-4; 

36 

55.8 

45-7 

37.3 

3o.8 

26.4 

24.1 

24.0 

26.3 

31.2 

38.8 

49-3| 

37 

57.3 

47.2 

38.8 

32.4 

28.0 

25.7 

20.7 

28.1 

33.o 

4o.6 

38 

58.8 

48.7 

4o.4 

34.o 

20.6 

27.4 

27.4 

29.8 

34.8 

42.4 

53*o 

39 

6o.3 

60.2 

41.9 

35.6 

3i.3 

29.0 

29.I 

3i.5 

36.5 

44.2 

54-9 

4o 

3261.  7 

3351.7 

3443.5 

3537.i 

3632.8 

373o.7 

383o.8 

3933.2 

4o38.3 

4  1  46.  i 

4256.  7 

4i 

63.2 

53.2 

45.o 

38.7 

34.4 

32.3 

32.4 

35.o 

4o.i 

47-9 

58.6! 

42 

64-7 

54.7 

46.6 

4o.3 

36.i 

34.o 

34.i 

36.7 

4i.8 

6o.5 

43 

66.2 

56.2 

48.i 

41.9 

37.7 

35.6 

35.8 

38.4 

43.6 

5i.5 

6a.3 

44 

67.7 

57.8 

49-7 

43.5 

39.3 

37.3 

37.5 

40.2 

45.4 

53.4 

64-2 

45 

69.2 

59.3 

5.1.9 

45.o 

40.9 

38.9 

39.2 

4i-9 

47.2 

55.2 

66.1 

46 

70.7 

60.8 

52.8 

46.6 

42.5 

4o.6 

4o.9 

43.6 

49.o 

57.0   » 

47 

72.1 

62.3 

54.3 

48.2 

44-1 

42.2 

42.6 

45-4 

5o.7 

58.8;  t 

48 

73.6 

63.8 

55.8 

49.8 

45.8 

43.9 

44-3 

47.1 

52.5 

60.7 

7'-7 

49 

75.i 

65.4 

57-4 

5i.4 

474 

45.5 

46.o 

48.8 

54.3 

62.5 

73.6 

5o 

3276.6 

3366.9 

3458.9 

3553.o 

3649-0 

3747.2 

3847-7 

395o.6 

4o56.i 

4i64.3J4a75.5j 

5i 

78.1 

68.4 

6o.5 

54-6 

5o.6 

48.8 

49.4 

52.3 

57.8 

66.1 

77-4 

5a 

79.6 

69.9 

62.0 

56.i 

52.2 

5o.5 

5i.i 

54.o 

596 

68.0 

79.2 

53 

81.1 

63.6 

57.7 

53.8 

52.1 

52.8 

55.8 

61.4 

69.8 

81.1 

54 

82.6 

7^0 

65.2 

59.3 

55.5 

53.8 

54-4 

57.5 

63.2 

71.6 

83.o 

55 

84.i 

74.5 

66.7 

60.9 

57.i 

55.5 

56.i 

59.3 

65.o 

73.5 

84-9 

56 

85.6 

76.0 

68.3 

62.5 

58.7 

5.7.i 

57.8 

61.0 

66.8 

75.3 

86.8 

1  57 

87.1 

77.5 

69.8 

64.0 

6o.3 

56.8 

59.5 

62.  7 

68.5 

77.1 

88.6 

58 

88.5 

79.0 

7r.4 

65.6 

61.9 

6o.4 

61.2 

64-5 

7o.3 

79.0 

9o.5 

90.0 

80.6 

73.o 

67.2 

63.6 

62.1 

62.  9 

66.2 

72.1 

80.8 

92.4 

5o 
5 1 

52 

53 
54 
55 
56 

57 
58 
59 


MERILIONAL    TARTS. 


i 
LATITUDE. 

Win. 

58° 

59° 

60° 

61° 

62° 

63° 

64° 

65° 

66° 

67° 

68° 

Min. 

0 

4294.3 

4409.1 

4527.4 

4649.2 

4775.o 

4904.9 

5o39.4 

5178.8 

5323.5 

5474.0 

563o.8 

0 

I 

96.2 

i  i.i 

29.4 

5i.3 

77.1 

07.1 

4i.7 

81.2 

26.0 

76.6 

33.5 

I 

2 

98.1 

i3.o 

3i.4 

53.4 

79.3 

09.4 

44-0 

83.5 

28.4 

79.1 

36.2 

2 

3 

43oo.o 

i5.o 

33.4 

55.4 

81.4 

11.6 

46.3 

85.9 

30.9 

81.7 

38.8 

3 

4 

oi.g 

16.9 

35-4 

57.5 

83.5 

i3.8 

48.6 

88.3 

33.4 

84.3 

4i.5 

4 

5 

03.7 

18.9 

87.41  59.6 

85.6 

16.0 

5o-9 

90.7 

35.8 

86.9 

44.2 

5 

6 

o5,6 

20.8 

394l  61.6 

87.8 

18.2 

53.i 

98.1 

38.3 

89.4 

46.9 

e 

7 

o7.5 

22.8 

4i-4 

63.7 

89.9 

20.4 

55.4 

95.4 

4o.8 

92.0 

49.6 

7 

8 

09.4 

24.7 

43.4 

65.8 

92.l|   22.6 

57.7 

97.8 

43.2 

94.5 

52.3 

8 

9 

n.3 

26.7 

45.4 

67.8 

94.2 

24.8 

60.0 

52OO.2 

45-7 

97.1 

54.9 

9 

10 

43i3.2 

4428.6 

4547-4 

4669.9 

4796.3 

4927.0 

5o62.3 

52O2.6 

5348.2 

5499.7 

5657.6 

10 

ii 

i5.i 

3o.6 

49.4 

72.0 

98.5 

29.2 

64-6 

04.9 

50.7 

55o2.3 

6o.3 

ii 

12 

17.0 

32.5 

5i.4 

74-i 

4800.6 

3i.5 

66.9 

07.3 

53.i 

o4-9 

63.o 

12 

i3 

18.9 

34.5 

53.5 

76.1 

02.8 

33.7 

69.2 

09.7 

55.6 

07.4 

65.7 

i3 

i4 

20.8 

36.4 

55.5 

78.2 

o4-9 

SS.g 

7i.5 

I  2.  1 

58.i 

10.  0 

68.4 

i4 

i5 

22.7 

38.4 

57.5 

8o.3 

07.1 

38.i 

73.8 

i4.5 

60.6 

12.6 

71.1 

i5 

16 

24.6 

4o.3 

59.5 

82.4 

09.2 

4o.3 

76.1 

16.9 

63.i 

15.2 

73.8 

16 

ll 

26.5 

42.3 

6i.5 

84.5 

n.4 

42.6 

78.4 

[9.3 

65.6 

17.8 

76.5 

'**? 

iS 

28.4 

44-2 

63.5 

86.5 

i3.5 

44-8 

80.7 

21.6 

68.0 

20.4 

•79.2 

18 

'9 

3o.3 

46.2 

65.6 

88.6 

16.7 

47.o 

83.o 

24.0 

70.5 

23.0 

81.9 

'9 

20 

4332.2 

4448.2 

4567.6 

4690.7 

4817.8 

4949.2 

5o85.3 

5226.4 

5373.o 

5525.6 

5684-6 

20 

21 

34.1 

5o.i 

69.6 

92.8 

20.  o 

5i.5 

87.6 

28.8 

75.5 

28.1 

87.3 

21 

22 

36.o 

52.1 

71.6 

94.9 

22.1 

53.7 

90.0 

3l.2 

78.0 

30.7 

90.0 

22 

23 

37.9 

54.1 

73.6 

97.0 

24.3 

55.9 

92.3 

33.6 

8o.5 

33.3 

92.7 

23 

24 

39.8 

56.o 

75.7 

99.1 

26.4 

58.2 

94.6 

36.o 

83.o 

35.9 

95.5 

24 

25 

4i.8 

58.o 

77-7 

4701.1 

28.6 

60.4 

96.9 

38.4 

85.5 

38.6 

98.2 

25 

26 

43.7 

60.0 

79-7 

03.2 

3o.8 

62.6 

99.2 

4o.8 

88.0 

41.2 

5700.9 

26 

27 

45.6 

61.9 

81.7 

o5.3 

32.9 

64.9 

5ioi.5 

43.2 

90.5 

43.8 

o3.6 

2-7 

28 

47.5 

63.9 

83.8 

07.4 

35.i 

67.1 

o3.8 

45.7 

93.0 

46.4 

o6.3 

28 

29 

49.4 

65.9 

85.8 

o9.5 

37.3 

69.4 

06.2 

48.i 

95.5 

49.0 

09.1 

29 

3o 

435i.3 

4467.8 

4587.8 

4711.6 

4839.4 

4971.6 

5io8.5 

D250.5 

5398.0 

555i.6 

6711.8 

3o 

3i 

53.2 

69.8 

89.9 

i3.7 

4i-6 

73.8 

10.8 

52.9 

5400.5 

54.2 

i4.5 

3i 

32 

55.i 

71.8 

91.9 

i5.8 

43.8 

76.1 

i3.i 

55.3 

o3.o 

56.8 

i7.3 

32 

33 

57.i 

73.7 

93.9 

i7.9 

45.9 

78.3 

i5.5 

57.7 

o5.5 

59.4 

20.O 

33 

34 

Sg.o 

75.7 

96.0 

20.0 

48.i 

80.6 

17.8 

60.  i 

08.1 

62.1 

22-7 

34 

35 

60.9 

77-7 

98.0 

22.1 

5o.3 

82.8 

2O.I 

62.6 

10.6 

64-7 

25.5 

35 

36 

62.8 

79-7 

46oo.  o 

24.2 

52.4 

85.i 

22.4 

65.o 

i3.i 

67.3 

28.2 

36 

3? 

64-7 

81.6 

02.  i 

26.3 

54-6 

87.3 

24.8 

67.4 

i5.6 

69.9 

3o-9 

37 

38 

66.7 

83.6 

o4-i 

cS.4 

56.8 

89.6 

27.1 

69.8 

18.1 

72.6 

33.7 

38 

39 

68.6 

85.6 

06.1 

3o.5 

59.0 

91.8 

29.4 

72.2 

20.6 

75.2 

36.4 

39 

4o 

4370.5 

4487.6 

4608.2 

4732.6 

486i.i 

4994.1 

5i3i.8 

5274.7 

5423.2 

5577.8 

5739.2 

4o 

4i 

72.4 

89.6 

10.2 

34.7 

63.3 

96.3 

34.i 

77.1 

25.7 

80.4 

41.9 

4i 

42 

74-3 

9i.5 

12.3 

36.8 

65.5 

98.6 

36.5 

79.5 

28.2 

83.i 

44.7 

42 

43 

76.3 

93.5 

i4.3 

39.0 

67.7 

5ooo.8 

38.8 

82.0 

3o.8 

85.7 

47.5 

43 

44 

78.2 

95.5 

16.4 

4i.i 

69.9 

o3.i 

4i.i 

84-4 

33.3 

88.4 

50.2 

44 

45 

80.  i 

97.5 

18.4 

43.2 

72.0 

o5.4 

43.5 

86.8 

35.8 

91.0 

52.9 

45 

46 

82.1 

99.5 

20.  5 

45.3 

74.2 

07.6 

45.8 

89.3 

38.4 

93.6 

55.7 

46 

4? 

84.0 

45oi.5 

22.5 

474 

76.4 

09.9 

48.2 

91.7 

40.9 

96.3 

58.5 

47 

48 

85.9 

o3.4 

24.6 

49-5 

78.6 

12.2 

5o.5 

94.1 

43.4 

98.9 

61.2 

48 

49 

87.8 

o5.4 

26.6 

5i.6 

80.8 

l44 

52.9 

96.6 

46.o 

56oi.6 

64.o 

49 

5o 

4389.8 

4507.4 

4628.7 

4753.7 

4883.o'5oi6.7 

5i55.2 

5299.0 

5448.5 

56o4.2 

5766.8 

5o 

5i 

91.7 

09.4 

3o.7 

55.9 

85.2 

18.9 

57.6 

53oi.5 

5i.o 

06.9 

69.5 

5i 

52 

93.6 

11.4 

32.8 

58.o 

87.4 

21.2 

59.9 

o3.9 

53.6 

09.5 

72.3 

52 

53 

95.6 

i3.4 

34.8 

60.  i 

89.5 

23.5 

62.3 

o6.3 

56.i 

12.2 

$5.i 

53 

54 

97.5 

1  5.4 

36.9 

62.2 

91.  7 

25.8 

64.6 

08.8 

58.7 

i4.8 

77-9 

54 

55 

99.4 

17-4 

39.0 

64-4 

93.9 

28.0 

67.0 

II.  2 

61.2 

i7.5 

80.6 

55 

56 

44oi-4 

i94 

4i.o 

66.5 

96.1 

3o.3 

69.4 

i3.7 

63.8 

2O.2 

83.4 

56 

57 

o3.3 

21.4 

43.o 

68.6 

98.3 

32.6 

71.7 

16.1 

66.3 

22.9 

86.2 

57 

'  58 

o5.3 

23.4 

45.i 

7°-7 

4900.5 

34-9 

74.i 

18.6 

68.9 

25.5 

89.0 

58 

59 

07.2 

25.4 

47-2 

72-9 

02.7 

37.i 

76.4 

21.  1 

7i.4 

28.2 

91.8 

59 

148 


MERIDIONAL    PARTS. 


LATITUDE. 

Min. 

69° 

70° 

71° 

72° 

73° 

74° 

75° 

76° 

77° 

78° 

79° 

MiB. 

O 

6794.6 

5965.9 

6145.7 

6334-8 

6534.4 

6745.7 

697o.3 

7210.1 

7467.2  7744-6 

8045.7 

O 

I 

97-4 

68.8 

48.8 

38.i 

37.8 

49.4 

74.2 

14.2 

71.7 

49.4 

5i.o 

I 

2 

58oo.i 

71.8 

5!.9 

4i.3 

4i.3 

53.o 

78.1 

E'8.3 

76.1 

54.2 

56.2 

2 

3 

02.9 

74-7 

54-9 

44-6 

44.7 

56.6 

81.9 

22.5 

80.6 

59.o 

6i.5 

3 

4 

o5.7 

77.6 

58.o 

4/-8 

48.i 

6o.3 

85.8 

26.6 

85.o 

63.9 

66.7 

4 

5 

o8.5 

80.6 

61.1 

5i.i 

5i.6 

63.9 

89.7 

3o.8 

89.5 

68.7 

72.0 

5 

6 

ii.  3 

83.5 

64.2 

54.3 

55.o 

67.6 

93.6 

35.o 

94.0 

73.5 

77.3 

6 

7 

14.2 

86.4 

67.3 

57.6 

58.5 

71.2 

97.5 

3o.i 

98.5 

78.4 

82.6 

7 

8 

17.0 

89.4 

70.4 

60.8 

61.9 

74.9 

7001.4 

43.3 

75o3.o 

83.3 

87-9 

8 

9 

19.8 

92.3 

73.5 

64.1 

65.3 

78.5 

o5.3 

47.5 

07.4 

88.1 

93.2 

9 

10 

5822.6 

5995.3 

6176.6 

6367.4 

6568.8 

6782.2 

7009.2 

7251.7 

7511.9 

7793.o 

8o98.5 

10 

ii 

25.4 

98.2 

79-7 

70.6 

72.3 

85.9 

1  3.  i 

55.8 

i6.5 

97-9 

8io3.8 

ii 

12 

28.2 

6001.2 

82.8 

73.9 

75.7 

89.6 

17.0 

60.0 

21.0 

7802.8 

09.2 

12 

i3 

3i.o 

o4-i 

85.9 

77.2 

79.2 

93.2 

20.9 

64.2 

25.5 

07.7 

i4.5 

i3 

i4 

33.8 

07.1 

89.0 

80.4 

82.6 

96.9 

24.8 

68.4 

3o.o 

12.6 

i9.9 

i4 

i5 

36.7 

IO.O 

92.1 

83.7 

86.1 

6800.6 

28.8 

72.6 

34-5 

i7.5 

25.2 

i5 

16 

Sg.S 

i3.o   55.2 

87.0 

89.6 

o4-3 

32.7 

76.8 

39.1 

22.4 

3o.6 

16 

*7 

42.3 

16.0 

98.3 

90.3 

93.0 

08.0 

36.6 

81.1 

43.6 

27.3 

36.0 

*7 

18 

45.i 

18.9 

6201.4 

93.6 

96.5 

u.6 

4o.6 

85.3 

48.i 

32.2 

4i.3 

18 

'9 

48.o 

21.9 

o4-5 

96.9 

6600.0 

i5.4 

44.5 

89.5 

62.7 

37.2 

46.7 

X9 

20 

585o.8 

6024.9 

6207.7 

64oo.2 

66o3.5 

6819.1 

7o48.5 

7293.7 

7557.3 

7842.i 

8162.1 

20 

21 

53.6 

27.8 

10.8 

o3.4 

o7.o 

22.8 

52.4 

98.o 

61.8 

47.i 

57.5 

21 

22 

56.5 

3o.8 

i3.9 

06.7 

io.5 

26.5 

56.4 

7302.2 

66.4 

52.  0 

62.9 

22 

23 

59.3 

33.8 

17.0 

10.  1 

i4«o 

30.2 

6o.3 

06.4 

71.0 

57.o 

68.4 

23 

24 

62.2 

36.8 

20.2 

i3.4 

i7.5 

33.9 

64.3 

10.7 

75.5 

6i.9 

73.8 

24 

25 

65.o 

39.8 

23.3 

16.7 

21.  0 

37.6 

68.3 

1  5.0 

80.1 

66.9 

79.2 

25 

16 

67.8 

42.7 

26.5 

20.  o 

24-5 

4i.3 

72.2 

19.2 

84-7 

71.9 

84-7 

26 

27 

70.7 

45.7 

29.6 

23.3 

28.0 

45.i 

76.2 

23.5 

89.3 

76.9 

9o.i 

27 

28 

73.5 

48.7 

32.7 

26.6 

3i.5 

48.8 

80.2 

27.7 

93.9 

81.9 

95.6 

28 

29 

76.4 

5i.7 

35.9 

29.9 

35.o 

52.5 

84.2 

32.0 

98.5 

86.9 

8201.1 

29 

3o 

5379.2 

6o54-7 

6239.0 

6433.3 

6638.5 

6856.3 

7o88.2 

7336.3 

7603.2 

789T-9 

8206.6 

3o 

3i 

82.1 

57.7 

42.2 

36.6 

42.1 

60.0 

92.2 

4o.6 

07.8 

96.9 

12.  1 

3! 

32 

85.o 

60.7 

45.3 

39.9 

45.6 

63.8 

96.2 

44.9 

12.4 

7902.O 

i7.6 

32 

33 

87.8 

63.7 

48.5 

43.2 

49.1 

67.5 

7IOO.2 

49.2 

17.0 

o7.o 

23.1 

33 

34 

90.7 

66.7 

5i.7 

46.6 

52.6 

7i.3 

04.2 

53.5 

21.7 

12.  0 

28.6 

34 

35 

93.6 

69.7 

54.8 

49.9 

56.2 

75.o 

08.2 

57.8 

26.3 

I7.I 

34.1 

35 

36 

96.4 

72.7 

58.o 

53.3 

59.7 

78.8 

12.2 

Sa.l 

3i.o 

22.1 

39.7 

36 

3? 

99.3 

75.7 

61.2 

56.6 

63.3 

82.6 

i6.3 

66.4 

35.6 

27.2 

45.2 

3? 

38 

5902.2 

78.8 

64-3 

60.0 

66.8 

86.3 

20.3 

70.7 

4o.3 

32.3 

5o.8 

38 

39 

o5.o 

81.8 

67.5 

63.3 

70.4 

90.1 

24.3 

75.i 

45.o 

37.3 

56.3 

39 

4o 

5907.9 

6o84.8 

6270.7 

6466.7 

6673.9 

6893.9 

7i28.4 

7379-4 

7649.7 

7942.4 

8261.9 

4o 

4i 

10.8 

87.8 

73.9 

70.0 

77*4 

97-7 

32.4 

83.7 

54.3 

47.5 

67.5 

4i 

42 

13.7 

90.8 

77.1 

73.4 

81.0 

6ooi.5 

36.4 

88.1 

59.o 

52.6 

73.i 

4'2 

43 

16.6 

93.9 

80.2 

76.7 

84-6 

o5.3 

4o.5 

92.4 

63.  7 

57.7 

78.6 

43 

44 

19.4 

96.9 

83.4 

80.1 

88.2 

09.1 

44-5 

96.8 

68.4 

62.8 

84-3 

44 

45 

22.3 

99.9 

86.6 

83.5 

91.7 

12.  9 

48.6 

74oi.i 

73.2 

68.0 

89.9 

45 

46 

25.2 

6io3.o 

89.8 

86.9 

95.3 

16.7 

52.7 

o5.5 

77-9 

73.i 

95.5 

46 

47 

28.1 

06.0 

93.o 

90.2 

98.9 

20.  5 

56.7 

°9'9 

82.6 

78.2 

83oi.i 

47 

48 

3i.o 

09.0 

96.2 

93.6 

6702.5 

24-3 

60.8 

i4.3 

87.3 

83.4 

06.  7 

48 

49 

33.9 

12.  1 

99.4 

97.0 

06.  i 

28.1 

64-9 

18.6 

92.1 

38.5 

12.4 

49 

5o 

5986.8 

6n5.r 

6302.6 

65oo.4 

6709.7 

6931.9 

7169.  o 

7423.o 

7696.8 

7993.7 

83i8.i 

5o 

5i 

39.7 

18.2 

o5.8 

o3.8 

13.2 

35.7 

73.i 

27.4 

7701.5 

98-9 

23.8 

5i 

52 

42.6 

21.2 

09.1 

07.2 

16.8 

93.6 

77.2 

3i.8 

o6.3 

8oo4-o 

294 

52 

53 

45.5 

24.3 

12.3 

10.6 

20.4 

43.4 

81.2 

36.2 

IX.  I 

09.2 

35.i 

53 

54 

48.4 

27.3 

i5.5 

i4.o 

24.0 

47-2 

85.3 

4o.6 

i5.8 

i44 

4o.8 

54 

55 

5i.3 

3o.4 

18.7 

17-4 

27.7 

5i.i 

89.5 

45.1 

2o.6 

I9.6 

46.5 

55, 

56 

54-2 

33.4 

21.9 

20.8 

3i.3 

54.9 

93.6 

49.5 

25.4 

24.8 

52.2 

56 

57 

57.2 

36.5 

25.1 

24.2 

34-9 

58.8 

97-7 

53.9 

30.2 

3o.o 

58.0 

57 

58 

60.  i 

39.6 

28.4 

27.6 

38.5 

62.6 

7201.8 

58.3 

35.o 

35.2 

63.7 

58 

59 

63.o 

L_ 

42.6 

3i.6 

3  1.0 

4s.  j 

66.5 

o5.6 

62.8 

39.8 

4o.5 

69.4 

59 

CORRECTIONS    TO    MIDDLE    LATITUDE. 


149 


Mid. 

Lat. 

1° 

DIFFERENCE  OF  LATITUDE. 

Mid 
Lot. 

2^ 

3° 

4° 

5° 

6° 

7° 

8° 

9° 

10° 

11° 

12° 

13° 

14° 

15°J160 

IT 

18° 

19°  20° 

o 

i5 

/ 

0 

/ 

2 

/ 
3 

/ 
5 

/ 
7 

/ 
9 

12 

i5 

18 

22 

26 

3i 

36 

4i 

47 

52 

59 

65 

/ 
72 

0 

r5 

16 

0 

2 

3 

4 

6 

9 

II 

i4 

18 

21 

25 

3o 

34 

39 

44 

5o 

56 

62 

69 

iC 

17 

o 

2 

3 

4 

6 

8 

II 

i4 

ll 

2O 

24 

28 

33 

38 

43 

48 

54 

60 

66 

17 

18 

o 

3 

4 

6 

8 

IO 

i3 

16 

2O 

23 

27 

32 

36 

4i 

46 

52 

58 

64 

18 

J9 

o 

3 

4 

6 

8 

IO 

i3 

16 

*9 

22 

26 

3o 

35 

4o 

45 

5o 

56 

61 

T9 

20  ,  o 

2 

4 

5 

7 

IO 

12 

i5 

18 

22 

25 

29 

34 

38 

43 

48 

54 

60 

20 

21    o 

2   4 

5 

7 

9 

12 

i5 

18 

21 

25 

29 

33 

37 

42 

47 

62 

58 

21 

22  i  O 

2 

4 

5 

7 

9 

12 

i4 

ll 

21 

24 

28 

32 

36 

4i 

46 

5i 

56 

22 

23 

o 

2 

3 

5 

7 

9 

II 

i4 

17 

20 

23 

27 

3i 

35 

4o 

45 

5o 

55 

23 

24 

o 

2 

3 

5 

7 

9 

II 

i4 

16 

20 

23 

27 

3i 

35 

39 

44 

49 

54 

24 

25 

o 

2 

3 

5 

7 

9 

11 

i3 

16 

J9 

23 

2U 

3o 

34 

39 

43 

48 

53 

25 

26 

0 

2 

3 

5 

6 

8 

II 

i3 

16 

J9 

22 

26 

3o 

34 

38 

42 

47 

52 

26 

27 

o 

2 

3 

5 

6 

8 

II 

i3 

16 

J9 

22 

25 

29 

33 

37 

42 

47 

52 

27 

28 

o 

2 

3 

5 

6 

8 

10 

i3 

16 

18 

22 

25 

29 

33 

37 

4i 

46 

5i 

28 

29 

o 

2 

3 

5 

6 

8 

10 

i3 

i5 

18 

21 

25 

28 

32 

37 

4i 

46 

5i 

29 

3o 

o 

2 

3 

5 

6 

8 

IO 

i3 

i5 

18 

21 

25 

28 

32 

36 

4i 

45 

5o 

3o 

3i 

o 

2 

3 

5 

6 

8 

IO 

12 

i5 

18 

21 

24 

28 

32 

36 

4o 

45 

5o 

3i 

32 

0 

o 

2 

3 

4 

6 

8 

IO 

12 

i5 

18 

21 

24 

28 

32 

36 

4o 

45 

5o 

32 

33 

o 

o 

2 

3 

4 

6 

8 

10 

12 

i5 

18 

21 

24 

28 

32 

36 

4o 

45 

49 

33 

34 

o 

o 

2 

3 

4 

6 

8 

IO 

12 

i5 

18 

21 

24 

28 

32 

36 

4o 

45 

49 

34 

35 

o 

o 

2 

3 

4 

6 

8 

IO 

12 

i5 

18 

21 

24 

28 

32 

36 

4o 

45 

49 

35 

36 

0 

2 

3 

4 

6 

8 

IO 

12 

i5 

18 

21 

24 

28 

32 

36 

4o 

45 

49 

36 

3? 

o 

2 

3 

4 

6 

8 

10 

12 

i5 

18 

21 

24 

28 

32 

36 

4o 

45 

49 

37 

38 

o 

2 

3 

4 

6 

8 

10 

12 

i5 

18 

21 

24 

28 

32 

36 

4o 

45 

5o 

38 

39 

o 

2 

3 

4 

6 

8 

IO 

12 

i5 

18 

21 

24 

28 

32 

36 

4o 

45 

5o 

39 

4o 

o 

2 

3 

5 

6 

8 

IO 

13 

i5 

18 

21 

25 

28 

32 

36 

4i 

45 

5o 

4o 

4i  o 

2 

3 

5 

6 

8 

IO 

13 

i5 

18 

21 

25 

28 

32 

37 

4i 

46 

5ij  4i 

42   0 

2 

3 

5 

6 

8 

10 

i3 

i5 

18 

22 

25 

29 

33 

37 

4i 

46 

5i  i  42 

43  o 

2 

3 

5 

6 

8 

10 

i3 

16 

18 

22 

25 

29 

33 

37 

42 

46 

52'  43 

44  :  o 

2 

3 

5 

6 

8 

IO 

i3 

16 

J9 

22 

25 

29 

33 

38 

42 

47 

52 

44 

i 

45 

0 

2 

3 

5 

6 

8 

II 

i3 

16 

X9 

22 

26 

3o 

34 

38 

43 

48 

53 

45 

46 

o 

2 

3 

5 

6 

8 

II 

i3 

16 

*9 

22 

26 

3o 

34 

38 

43 

48 

53 

46 

47 

o 

2 

3 

5 

7 

9 

II 

i3 

16 

J9 

23 

26 

3o 

35 

39 

44 

49 

54 

47 

48 

o 

2 

3 

5 

7 

9 

II 

i4 

^7 

20 

23 

27 

3i 

35 

4o 

44 

5o 

55 

48 

49 

o 

2 

3 

5 

7 

9 

II 

i4 

17 

20 

23 

27 

3i 

36 

4o 

45 

5o 

56 

49 

5o 

0 

2 

4 

5 

7 

9 

II 

i4 

!7 

20 

24 

28 

32 

36 

4i 

46 

5i 

57 

5o 

5i 

o 

2 

4 

5 

7 

9 

12 

i4 

!7 

21 

24 

28 

32 

37 

42 

47 

52 

58 

5i 

52 

o 

2 

4 

5 

7 

9 

12 

i5 

18 

21 

25 

29 

33 

38 

43 

48 

53 

59 

52 

53 

o 

2 

4 

5 

7 

IO 

12 

i5 

18 

21 

25 

29 

34 

38 

43 

49 

54 

60 

53 

54 

o 

2 

4 

5 

7 

IO 

12 

i5 

18 

22 

26 

3o 

34 

39 

44 

5o 

56 

62 

54 

55 

o 

2 

4 

6 

8 

10 

13 

16 

J9 

22 

26 

3i 

35 

4o 

45 

5i 

57 

63 

55 

56 

o 

3 

4 

6 

8 

IO 

i3 

16 

r9 

23 

27 

3i 

36 

4i 

46 

52 

58 

65 

56 

57 

o 

3 

4 

6 

8 

IO 

i3 

16 

20 

24 

28 

32 

37 

42 

48 

54 

60 

66 

57 

.58 

o 

2 

3 

4 

6 

8 

II 

i4 

17 

2O 

24 

28 

33 

38 

43 

49 

55 

61 

68 

58 

59 

o 

2 

3 

4 

6 

8 

II 

i4 

ll 

21 

25 

29 

34 

39 

45 

5o 

57 

63 

70 

59 

60 

o 

2 

3 

4 

6 

9 

II 

i4 

18 

22 

26 

3o 

35 

4o 

46 

52 

58 

65 

72 

60 

6  1 

o 

2 

3 

5 

7 

9 

12 

i5 

18 

22 

26 

3i 

36 

42 

47 

53 

60 

67 

75 

61 

62 

0 

2 

3 

5 

7 

9 

12 

i5 

'9 

23 

27 

32 

37 

43 

49 

55 

62 

70 

77 

62 

63 

o 

2 

3 

5 

7 

10 

12 

16 

20 

24 

28 

33 

39 

44 

5i 

57 

64 

72 

80 

63 

64 

o 

2 

3 

5 

7 

10 

13 

16 

20 

24 

29 

34 

4o 

46 

52 

59 

67 

75 

83 

64 

05 

0 

2 

3 

5 

7 

IO 

i3 

J7 

21 

25 

3o 

36 

4i 

48 

54 

62 

69 

78 

86 

t>5 

66 

o 

2 

3 

5 

8 

II 

i4 

18 

32 

26 

32 

37 

43 

5o 

57 

64 

72 

81 

9° 

66 

67 

o 

2 

4 

6 

8 

II 

i4 

18 

23 

28 

33 

39 

45 

52 

59 

67 

76 

85 

94 

67 

68 

o 

2 

4 

6 

8 

12 

i5 

T9 

24 

29 

34 

4o 

47 

54 

62 

70 

79 

39 

99 

68 

69 

o 

2 

4 

6 

9 

12 

16 

20 

25 

3o 

36 

42 

49 

57 

65 

74 

83 

93 

io4;  69 

70 

o 

2 

4 

6 

9 

i3 

16 

21 

26 

32 

38 

44 

52 

60 

68 

78 

88 

98 

no 

7° 

7i 

0 

2 

4 

7 

10 

i3 

'7 

22 

27 

33 

4o 

47 

55 

63 

72 

82 

93 

io4 

116 

7' 

72 

o 

3 

5 

7 

10 

i4 

18 

23 

29 

35 

42 

49 

58 

67 

76,87 

98  -  1  1  1 

124 

7  P. 

150       LOGARITHMS    FOR    COMPUTING    COMPOUND    INTEREST. 


In  computing  compound  interest  for  long  periods  of  time, 
the  following  logarithms  to  more  than  six  places. 

it  is  necessary  to  have 

Number. 

Logarithm. 

Number. 

Logarithm. 

.0026 
.oo5o 

.0076 
.OIOO 
.0125 

.0160 
.0176 
.0200 
.0226 
.0260 
.0276 
.o3oo 
,o325 
.o35o 
.0876 
.o4oo 

.00108  438i3 

.00216  60618 

.oo324  5o548 
.oo432  i3738 
.00539  5o3i9 
.00646  60422 
.00753  44i79 
.00860  01718 
.00966  33167 
.01072  38654 
.01178  i83o5 
.01283  72247 
.01389  oo6o3 
.01494  03498 
.01598  8io54 
.01703  33393 

.o425 

.o45o 
.0476 
.o5oo 
.o525 
.o55o 
.o575 
.0600 
.0626 
.o65o 
.0675 
.0700 
.0726 
.0750 
.o775 
.0800 

.01807  6o636 
.01911  62904 
.O20i5  4o3i6 
.02118  92991 
.02222  2io45 
.O2325  24596 
.0242803760 

.O253o  58653 
.02632  89387 
.0273496078 
.02836  78837 
.02938  37777 
.oSoSg  73009 
.o3i4o  84643 
.o324i  72788 
.o3342  37555 

NUMBERS  OFTEN  USED  IN  CALCULATIONS. 

Circumference  c 
Surface  of  a  spb 
Area  of  a  circle 
Area  of  a  circle 
Capacity  of  ti  sp 
Capacity  of  a  sp 
i  —  3.1415926  . 

f  a  circle  to  diameter  i  ) 
ere  to  diameter  i  .  .  .  .  > 
to  radius  i                       S 

= 

3.i4i5926 

.7853982 
.5235988 
4.1887902 
0.3183099 
°.  2957795 
206264".  8 
.0174533 
.0002909 
.ooooo485 
.00000970 
.00001454 
.00001939 
.00002424 
.00002909 
.00003394 
.00003879 
.oooo4363 
2.7182818 
.4342945 
1296000 
864oo 
528o 

Logarithms. 

o.-497i5o 

9.895090 
9.718999 
0.622089 
9-5o285o 
i.758i23 
5.3i4425 
8.241877 
6.463726 
4.685575 
4.986605 
5.162696 
5.287635 
5.384545 
5.463726 
5.53o673 
5.588665 
5.639817 
0.434294 
9.637784 
6.ii26o5 
4-9365i4 
3.722634 

to  diameter  i 

here  to  diameter  i 



here  to  radius  i  



Arc  equal  to  rad 
Arc  equal  to  rad 
Length  of  i  deg 
Length  of  i  min 
Sine  of  i  seconc 
Sine  of  2  seconc 
Sine  of  3  seconc 
Sine  of  4  seconc 
Sine  of  5  seconc 
Sine  of  6  seconc 
Sine  of  7  seconc 
Sine  of  8  seconc 
Sine  of  9  seconc 
Base  of  Napier' 
Modulus  of  the 
36o  degrees  exp 
24  hours  expres 
Number  of  feet 

ius  expressed  in  degrees  . 

—  57 

ius  expressed  in  seconds 

ree  in  parts  of  radius 



ute  in  parts  of  radius  





s 

s  

Is  

3  

S  

S 

s  

s  

_ 

s  system  of  logarithms 



common  logarithms  

:  

ressed  in  seconds 



sed  in  seconds  



in  one  mile  

I 


THE    END. 


1C  49S87 


M289993 


Lr 


THE  UNIVERSITY  OF  CALIFORNIA  LIBRARY 


